Is cos^6(105)=((1+cos(210))/2)^3 ?

The answer to whether cos^6(105)=((1+cos(210))/2)^3 is True

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Popular Trigonometry >

prove cos^6(105)=((1+cos(210))/2)^3

Solution

prove

cos^{6} (10 5^{\circ}) = (\frac{1 + cos ( 21 0 ^{\circ} )}{2})^{3}

Solution

True

Solution steps

cos^{6} (10 5^{\circ}) = (\frac{1 + cos ( 21 0 ^{\circ} )}{2})^{3}

Manipulating left side

cos^{6} (10 5^{\circ})

Expand

cos^{6} (10 5^{\circ}) : \frac{13}{32} - \frac{15 3}{64}

cos^{6} (10 5^{\circ})

cos (10 5^{\circ}) = \frac{2 ( 1 - 3 )}{4}

cos (10 5^{\circ})

Rewrite using trig identities:

cos (6 0^{\circ}) cos (4 5^{\circ}) - sin (6 0^{\circ}) sin (4 5^{\circ})

cos (10 5^{\circ})

Write

cos (10 5^{\circ})

cos (6 0^{\circ} + 4 5^{\circ})

= cos (6 0^{\circ} + 4 5^{\circ})

Use the Angle Sum identity:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= cos (6 0^{\circ}) cos (4 5^{\circ}) - sin (6 0^{\circ}) sin (4 5^{\circ})

= cos (6 0^{\circ}) cos (4 5^{\circ}) - sin (6 0^{\circ}) sin (4 5^{\circ})

Use the following trivial identity:

cos (6 0^{\circ}) = \frac{1}{2}

cos (6 0^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{1}{2}

Use the following trivial identity:

cos (4 5^{\circ}) = \frac{2}{2}

cos (4 5^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

Use the following trivial identity:

sin (6 0^{\circ}) = \frac{3}{2}

sin (6 0^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

Use the following trivial identity:

sin (4 5^{\circ}) = \frac{2}{2}

sin (4 5^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

= \frac{1}{2} \cdot \frac{2}{2} - \frac{3}{2} \cdot \frac{2}{2}

Simplify

\frac{1}{2} \cdot \frac{2}{2} - \frac{3}{2} \cdot \frac{2}{2} : \frac{2 ( 1 - 3 )}{4}

\frac{1}{2} \cdot \frac{2}{2} - \frac{3}{2} \cdot \frac{2}{2}

Factor out common term

\frac{2}{2}

= \frac{2}{2} (\frac{1}{2} - \frac{3}{2})

\frac{1}{2} - \frac{3}{2} = \frac{1 - 3}{2}

\frac{1}{2} - \frac{3}{2}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 - 3}{2}

= \frac{2}{2} \cdot \frac{1 - 3}{2}

Multiply fractions:

\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}

= \frac{( 1 - 3 ) 2}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 ( 1 - 3 )}{4}

= \frac{2 ( 1 - 3 )}{4}

= (\frac{2 ( 1 - 3 )}{4})^{6}

Simplify

(\frac{2 ( 1 - 3 )}{4})^{6}

\frac{2 ( 1 - 3 )}{4} = \frac{2 - 6}{4}

\frac{2 ( 1 - 3 )}{4}

Expand

2 (1 - 3) : 2 - 6

2 (1 - 3)

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 1, c = 3

= 2 \cdot 1 - 23

= 1 \cdot 2 - 23

Simplify

1 \cdot 2 - 23 : 2 - 6

1 \cdot 2 - 23

1 \cdot 2 = 2

1 \cdot 2

Multiply:

1 \cdot 2 = 2

= 2

23 = 6

23

Apply radical rule:

a b = a \cdot b

23 = 2 \cdot 3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

= 2 - 6

= 2 - 6

= \frac{2 - 6}{4}

= (\frac{2 - 6}{4})^{6}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 2 - 6 ) ^{6}}{4 ^{6}}

(2 - 6)^{6} = 1664 - 9603

(2 - 6)^{6}

Apply binomial theorem:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = 2, b = - 6

= i = 0 \sum 6 (i 6) (2)^{(6 - i)} (- 6)^{i}

Expand summation

(i n) = \frac{n !}{i ! ( n - i ) !}

i = 0 : \frac{6 !}{0 ! ( 6 - 0 ) !} (2)^{6} (- 6)^{0}

i = 1 : \frac{6 !}{1 ! ( 6 - 1 ) !} (2)^{5} (- 6)^{1}

i = 2 : \frac{6 !}{2 ! ( 6 - 2 ) !} (2)^{4} (- 6)^{2}

i = 3 : \frac{6 !}{3 ! ( 6 - 3 ) !} (2)^{3} (- 6)^{3}

i = 4 : \frac{6 !}{4 ! ( 6 - 4 ) !} (2)^{2} (- 6)^{4}

i = 5 : \frac{6 !}{5 ! ( 6 - 5 ) !} (2)^{1} (- 6)^{5}

i = 6 : \frac{6 !}{6 ! ( 6 - 6 ) !} (2)^{0} (- 6)^{6}

= \frac{6 !}{0 ! ( 6 - 0 ) !} (2)^{6} (- 6)^{0} + \frac{6 !}{1 ! ( 6 - 1 ) !} (2)^{5} (- 6)^{1} + \frac{6 !}{2 ! ( 6 - 2 ) !} (2)^{4} (- 6)^{2} + \frac{6 !}{3 ! ( 6 - 3 ) !} (2)^{3} (- 6)^{3} + \frac{6 !}{4 ! ( 6 - 4 ) !} (2)^{2} (- 6)^{4} + \frac{6 !}{5 ! ( 6 - 5 ) !} (2)^{1} (- 6)^{5} + \frac{6 !}{6 ! ( 6 - 6 ) !} (2)^{0} (- 6)^{6}

= \frac{6 !}{0 ! ( 6 - 0 ) !} (2)^{6} (- 6)^{0} + \frac{6 !}{1 ! ( 6 - 1 ) !} (2)^{5} (- 6)^{1} + \frac{6 !}{2 ! ( 6 - 2 ) !} (2)^{4} (- 6)^{2} + \frac{6 !}{3 ! ( 6 - 3 ) !} (2)^{3} (- 6)^{3} + \frac{6 !}{4 ! ( 6 - 4 ) !} (2)^{2} (- 6)^{4} + \frac{6 !}{5 ! ( 6 - 5 ) !} (2)^{1} (- 6)^{5} + \frac{6 !}{6 ! ( 6 - 6 ) !} (2)^{0} (- 6)^{6}

\frac{6 !}{0 ! ( 6 - 0 ) !} (2)^{6} (- 6)^{0} = 8

\frac{6 !}{0 ! ( 6 - 0 ) !} (2)^{6} (- 6)^{0}

Apply rule

a^{0} = 1, a \neq = 0

(- 6)^{0} = 1

= 1 \cdot \frac{6 !}{0 ! ( 6 - 0 ) !} (2)^{6}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= 1 \cdot \frac{6 ! ( 2 ) ^{6}}{0 ! ( 6 - 0 ) !}

\frac{6 ! ( 2 ) ^{6}}{0 ! ( 6 - 0 ) !} = 8

\frac{6 ! ( 2 ) ^{6}}{0 ! ( 6 - 0 ) !}

0! (6 - 0)! = 6!

0! (6 - 0)!

Subtract the numbers:

6 - 0 = 6

= 0! \cdot 6!

Apply factorial rule:

0! = 1

= 1 \cdot 6!

Multiply:

1 \cdot 6! = 6!

= 6!

= \frac{6 ! ( 2 ) ^{6}}{6 !}

Cancel the common factor:

6!

= (2)^{6}

Apply radical rule:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{6}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 2^{\frac{1}{2} \cdot 6}

\frac{1}{2} \cdot 6 = 3

\frac{1}{2} \cdot 6

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 6}{2}

Multiply the numbers:

1 \cdot 6 = 6

= \frac{6}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3

= 2^{3}

2^{3} = 8

= 8

= 1 \cdot 8

Multiply the numbers:

1 \cdot 8 = 8

= 8

\frac{6 !}{1 ! ( 6 - 1 ) !} (2)^{5} (- 6)^{1} = - 483

\frac{6 !}{1 ! ( 6 - 1 ) !} (2)^{5} (- 6)^{1}

Apply rule

a^{1} = a

(- 6)^{1} = - 6

= \frac{6 !}{1 ! ( 6 - 1 ) !} (2)^{5} (- 6)

Remove parentheses:

(- a) = - a

= - \frac{6 !}{1 ! ( 6 - 1 ) !} (2)^{5} 6

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= - \frac{6 ! ( 2 ) ^{5} 6}{1 ! ( 6 - 1 ) !}

Subtract the numbers:

6 - 1 = 5

= - \frac{6 \cdot 6 ! ( 2 ) ^{5}}{1 ! \cdot 5 !}

Cancel the factorials:

\frac{n !}{( n - m ) !} = n \cdot (n - 1) \dots (n - m + 1), n > m

\frac{6 !}{5 !} = 6

= - \frac{6 6 ( 2 ) ^{5}}{1 !}

(2)^{5} = 2^{2} 2

(2)^{5}

Apply radical rule:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{5}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 2^{\frac{1}{2} \cdot 5}

\frac{1}{2} \cdot 5 = \frac{5}{2}

\frac{1}{2} \cdot 5

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 5}{2}

Multiply the numbers:

1 \cdot 5 = 5

= \frac{5}{2}

= 2^{\frac{5}{2}}

2^{\frac{5}{2}} = 2^{2} 2

2^{\frac{5}{2}}

2^{\frac{5}{2}} = 2^{2 + \frac{1}{2}}

= 2^{2 + \frac{1}{2}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

= 2^{2} \cdot 2^{\frac{1}{2}}

Refine

= 2^{2} 2

= 2^{2} 2

= - \frac{2 ^{2} \cdot 6 2 6}{1 !}

Simplify

6 \cdot 2^{2} 26 : 2^{3} \cdot 3 \cdot 23

6 \cdot 2^{2} 26

Factor integer

6 = 2 \cdot 3

= 2 \cdot 3 \cdot 2^{2} 26

Factor integer

6 = 2 \cdot 3

= 2 \cdot 3 \cdot 2^{2} 2 2 \cdot 3

Apply radical rule:

2 \cdot 3 = 23

= 2 \cdot 3 \cdot 2^{2} 223

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

2 \cdot 2^{2} = 2^{1 + 2}

= 3 \cdot 2^{1 + 2} 223

Add the numbers:

1 + 2 = 3

= 3 \cdot 2^{3} 223

Apply radical rule:

a a = a

22 = 2

= 2^{3} \cdot 3 \cdot 23

= - \frac{2 ^{3} \cdot 3 \cdot 2 3}{1 !}

Multiply the numbers:

3 \cdot 2 = 6

= - \frac{2 ^{3} \cdot 6 3}{1 !}

6 \cdot 2^{3} 3 = 483

6 \cdot 2^{3} 3

2^{3} = 8

= 6 \cdot 83

Multiply the numbers:

6 \cdot 8 = 48

= 483

= - \frac{48 3}{1 !}

Apply factorial rule:

n! = 1 \cdot 2 \cdot 3 \cdot \dots \cdot n

1! = 1

= - \frac{48 3}{1}

Apply rule

\frac{a}{1} = a

= - 483

\frac{6 !}{2 ! ( 6 - 2 ) !} (2)^{4} (- 6)^{2} = 360

\frac{6 !}{2 ! ( 6 - 2 ) !} (2)^{4} (- 6)^{2}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{6 ! ( 2 ) ^{4} ( - 6 ) ^{2}}{2 ! ( 6 - 2 ) !}

6! (2)^{4} (- 6)^{2} = 6! (2)^{4} (6)^{2}

6! (2)^{4} (- 6)^{2}

(- 6)^{2} = (6)^{2}

(- 6)^{2}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 6)^{2} = (6)^{2}

= (6)^{2}

= 6! (2)^{4} (6)^{2}

= \frac{6 ! ( 2 ) ^{4} ( 6 ) ^{2}}{2 ! ( 6 - 2 ) !}

Subtract the numbers:

6 - 2 = 4

= \frac{6 ! ( 2 ) ^{4} ( 6 ) ^{2}}{2 ! \cdot 4 !}

Cancel the factorials:

\frac{n !}{( n - m ) !} = n \cdot (n - 1) \dots (n - m + 1), n > m

\frac{6 !}{4 !} = 6 \cdot 5

= \frac{6 \cdot 5 ( 2 ) ^{4} ( 6 ) ^{2}}{2 !}

Refine

= \frac{30 ( 2 ) ^{4} ( 6 ) ^{2}}{2 !}

(2)^{4} = 2^{2}

(2)^{4}

Apply radical rule:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{4}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 2^{\frac{1}{2} \cdot 4}

\frac{1}{2} \cdot 4 = 2

\frac{1}{2} \cdot 4

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 4}{2}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2

= 2^{2}

= \frac{2 ^{2} \cdot 30 ( 6 ) ^{2}}{2 !}

(6)^{2} = 6

(6)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (6^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 6^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 6

= \frac{2 ^{2} \cdot 30 \cdot 6}{2 !}

Multiply the numbers:

30 \cdot 6 = 180

= \frac{2 ^{2} \cdot 180}{2 !}

180 \cdot 2^{2} = 720

180 \cdot 2^{2}

2^{2} = 4

= 180 \cdot 4

Multiply the numbers:

180 \cdot 4 = 720

= 720

= \frac{720}{2 !}

2! = 2

2!

Apply factorial rule:

n! = 1 \cdot 2 \cdot 3 \cdot \dots \cdot n

2! = 1 \cdot 2

= 1 \cdot 2

Multiply the numbers:

1 \cdot 2 = 2

= 2

= \frac{720}{2}

Divide the numbers:

\frac{720}{2} = 360

= 360

\frac{6 !}{3 ! ( 6 - 3 ) !} (2)^{3} (- 6)^{3} = - 4803

\frac{6 !}{3 ! ( 6 - 3 ) !} (2)^{3} (- 6)^{3}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{6 ! ( 2 ) ^{3} ( - 6 ) ^{3}}{3 ! ( 6 - 3 ) !}

6! (2)^{3} (- 6)^{3} = - 6! (23)^{3}

6! (2)^{3} (- 6)^{3}

(- 6)^{3} = - (6)^{3}

(- 6)^{3}

Apply exponent rule:

(- a)^{n} = - a^{n},

n

is odd

(- 6)^{3} = - (6)^{3}

= - (6)^{3}

= 6! (2)^{3} (- (6)^{3})

Remove parentheses:

(- a) = - a

= - 6! (2)^{3} (6)^{3}

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

(2)^{3} (6)^{3} = (26)^{3}

= - 6! (26)^{3}

26 = 23

26

Factor integer

6 = 2 \cdot 3

= 2 2 \cdot 3

Apply radical rule:

2 \cdot 3 = 23

= 223

Apply radical rule:

a a = a

22 = 2

= 23

= - 6! (23)^{3}

= \frac{- 6 ! ( 2 3 ) ^{3}}{3 ! ( 6 - 3 ) !}

Subtract the numbers:

6 - 3 = 3

= \frac{- 6 ! ( 2 3 ) ^{3}}{3 ! \cdot 3 !}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{6 ! ( 2 3 ) ^{3}}{3 ! \cdot 3 !}

Cancel

\frac{6 ! ( 2 3 ) ^{3}}{3 ! \cdot 3 !} : \frac{120 ( 2 3 ) ^{3}}{3 !}

\frac{6 ! ( 2 3 ) ^{3}}{3 ! \cdot 3 !}

Cancel the factorials:

\frac{n !}{( n - m ) !} = n \cdot (n - 1) \dots (n - m + 1), n > m

\frac{6 !}{3 !} = 6 \cdot 5 \cdot 4

= \frac{6 \cdot 5 \cdot 4 ( 2 3 ) ^{3}}{3 !}

Refine

= \frac{120 ( 2 3 ) ^{3}}{3 !}

= - \frac{120 ( 2 3 ) ^{3}}{3 !}

(23)^{3} = 2^{3} \cdot 33

(23)^{3}

Apply exponent rule:

(a \cdot b)^{n} = a^{n} b^{n}

= 2^{3} (3)^{3}

(3)^{3} : 3^{\frac{3}{2}}

Apply radical rule:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{3}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 3}

\frac{1}{2} \cdot 3 = \frac{3}{2}

\frac{1}{2} \cdot 3

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 3}{2}

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3}{2}

= 3^{\frac{3}{2}}

= 2^{3} \cdot 3^{\frac{3}{2}}

3^{\frac{3}{2}} = 33

3^{\frac{3}{2}}

3^{\frac{3}{2}} = 3^{1 + \frac{1}{2}}

= 3^{1 + \frac{1}{2}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

= 3^{1} \cdot 3^{\frac{1}{2}}

Refine

= 33

= 2^{3} \cdot 33

= - \frac{2 ^{3} \cdot 120 \cdot 3 3}{3 !}

Multiply the numbers:

120 \cdot 3 = 360

= - \frac{2 ^{3} \cdot 360 3}{3 !}

360 \cdot 2^{3} 3 = 28803

360 \cdot 2^{3} 3

2^{3} = 8

= 360 \cdot 83

Multiply the numbers:

360 \cdot 8 = 2880

= 28803

= - \frac{2880 3}{3 !}

3! = 6

3!

Apply factorial rule:

n! = 1 \cdot 2 \cdot 3 \cdot \dots \cdot n

3! = 1 \cdot 2 \cdot 3

= 1 \cdot 2 \cdot 3

Multiply the numbers:

1 \cdot 2 \cdot 3 = 6

= 6

= - \frac{2880 3}{6}

Divide the numbers:

\frac{2880}{6} = 480

= - 4803

\frac{6 !}{4 ! ( 6 - 4 ) !} (2)^{2} (- 6)^{4} = 1080

\frac{6 !}{4 ! ( 6 - 4 ) !} (2)^{2} (- 6)^{4}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{6 ! ( 2 ) ^{2} ( - 6 ) ^{4}}{4 ! ( 6 - 4 ) !}

6! (2)^{2} (- 6)^{4} = 6! (6)^{4} (2)^{2}

6! (2)^{2} (- 6)^{4}

(- 6)^{4} = (6)^{4}

(- 6)^{4}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 6)^{4} = (6)^{4}

= (6)^{4}

= 6! (6)^{4} (2)^{2}

= \frac{6 ! ( 6 ) ^{4} ( 2 ) ^{2}}{4 ! ( 6 - 4 ) !}

Subtract the numbers:

6 - 4 = 2

= \frac{6 ! ( 6 ) ^{4} ( 2 ) ^{2}}{4 ! \cdot 2 !}

Cancel the factorials:

\frac{n !}{( n - m ) !} = n \cdot (n - 1) \dots (n - m + 1), n > m

\frac{6 !}{4 !} = 6 \cdot 5

= \frac{6 \cdot 5 ( 6 ) ^{4} ( 2 ) ^{2}}{2 !}

Refine

= \frac{30 ( 6 ) ^{4} ( 2 ) ^{2}}{2 !}

(2)^{2} = 2

(2)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (2^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 2^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2

= \frac{30 \cdot 2 ( 6 ) ^{4}}{2 !}

(6)^{4} = 6^{2}

(6)^{4}

Apply radical rule:

a = a^{\frac{1}{2}}

= (6^{\frac{1}{2}})^{4}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 6^{\frac{1}{2} \cdot 4}

\frac{1}{2} \cdot 4 = 2

\frac{1}{2} \cdot 4

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 4}{2}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2

= 6^{2}

= \frac{6 ^{2} \cdot 30 \cdot 2}{2 !}

Multiply the numbers:

30 \cdot 2 = 60

= \frac{6 ^{2} \cdot 60}{2 !}

60 \cdot 6^{2} = 2160

60 \cdot 6^{2}

6^{2} = 36

= 60 \cdot 36

Multiply the numbers:

60 \cdot 36 = 2160

= 2160

= \frac{2160}{2 !}

2! = 2

2!

Apply factorial rule:

n! = 1 \cdot 2 \cdot 3 \cdot \dots \cdot n

2! = 1 \cdot 2

= 1 \cdot 2

Multiply the numbers:

1 \cdot 2 = 2

= 2

= \frac{2160}{2}

Divide the numbers:

\frac{2160}{2} = 1080

= 1080

\frac{6 !}{5 ! ( 6 - 5 ) !} (2)^{1} (- 6)^{5} = - 4323

\frac{6 !}{5 ! ( 6 - 5 ) !} (2)^{1} (- 6)^{5}

Apply rule

a^{1} = a

(2)^{1} = 2

= 2 \frac{6 !}{5 ! ( 6 - 5 ) !} (- 6)^{5}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{6 ! 2 ( - 6 ) ^{5}}{5 ! ( 6 - 5 ) !}

6! 2 (- 6)^{5} = - 2 \cdot 6! (6)^{5}

6! 2 (- 6)^{5}

(- 6)^{5} = - (6)^{5}

(- 6)^{5}

Apply exponent rule:

(- a)^{n} = - a^{n},

n

is odd

(- 6)^{5} = - (6)^{5}

= - (6)^{5}

= 2 \cdot 6! (- (6)^{5})

Remove parentheses:

(- a) = - a

= - 6! 2 (6)^{5}

= \frac{- 2 \cdot 6 ! ( 6 ) ^{5}}{5 ! ( 6 - 5 ) !}

Subtract the numbers:

6 - 5 = 1

= \frac{- 2 \cdot 6 ! ( 6 ) ^{5}}{5 ! \cdot 1 !}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{6 ! 2 ( 6 ) ^{5}}{5 ! \cdot 1 !}

Cancel the factorials:

\frac{n !}{( n - m ) !} = n \cdot (n - 1) \dots (n - m + 1), n > m

\frac{6 !}{5 !} = 6

= - \frac{6 2 ( 6 ) ^{5}}{1 !}

(6)^{5} = 6^{2} 6

(6)^{5}

Apply radical rule:

a = a^{\frac{1}{2}}

= (6^{\frac{1}{2}})^{5}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 6^{\frac{1}{2} \cdot 5}

\frac{1}{2} \cdot 5 = \frac{5}{2}

\frac{1}{2} \cdot 5

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 5}{2}

Multiply the numbers:

1 \cdot 5 = 5

= \frac{5}{2}

= 6^{\frac{5}{2}}

6^{\frac{5}{2}} = 6^{2} 6

6^{\frac{5}{2}}

6^{\frac{5}{2}} = 6^{2 + \frac{1}{2}}

= 6^{2 + \frac{1}{2}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

= 6^{2} \cdot 6^{\frac{1}{2}}

Refine

= 6^{2} 6

= 6^{2} 6

= - \frac{6 ^{2} \cdot 6 2 6}{1 !}

62 \cdot 6^{2} 6 = 6^{3} 26

62 \cdot 6^{2} 6

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

6 \cdot 6^{2} = 6^{1 + 2}

= 2 \cdot 6^{1 + 2} 6

Add the numbers:

1 + 2 = 3

= 2 \cdot 6^{3} 6

= - \frac{6 ^{3} 2 6}{1 !}

Simplify

2 \cdot 6^{3} 6 : 2^{4} \cdot 3^{3} 3

2 \cdot 6^{3} 6

Factor integer

6 = 2 \cdot 3

= 2 (2 \cdot 3)^{3} 6

Apply exponent rule:

(ab)^{c} = a^{c} b^{c}

(2 \cdot 3)^{3} = 2^{3} \cdot 3^{3}

= 2 \cdot 2^{3} \cdot 3^{3} 6

Factor integer

6 = 2 \cdot 3

= 2 \cdot 2^{3} \cdot 3^{3} 2 \cdot 3

Apply radical rule:

2 \cdot 3 = 23

= 2 \cdot 2^{3} \cdot 3^{3} 23

Apply radical rule:

a a = a

22 = 2

= 2^{3} \cdot 3^{3} \cdot 23

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

2^{3} \cdot 2 = 2^{3 + 1}

= 3^{3} \cdot 2^{3 + 1} 3

Add the numbers:

3 + 1 = 4

= 3^{3} \cdot 2^{4} 3

= - \frac{2 ^{4} \cdot 3 ^{3} 3}{1 !}

3^{3} \cdot 2^{4} 3 = 4323

3^{3} \cdot 2^{4} 3

3^{3} = 27

= 2^{4} \cdot 273

2^{4} = 16

= 27 \cdot 163

Multiply the numbers:

27 \cdot 16 = 432

= 4323

= - \frac{432 3}{1 !}

Apply factorial rule:

n! = 1 \cdot 2 \cdot 3 \cdot \dots \cdot n

1! = 1

= - \frac{432 3}{1}

Apply rule

\frac{a}{1} = a

= - 4323

\frac{6 !}{6 ! ( 6 - 6 ) !} (2)^{0} (- 6)^{6} = 216

\frac{6 !}{6 ! ( 6 - 6 ) !} (2)^{0} (- 6)^{6}

Apply rule

a^{0} = 1, a \neq = 0

(2)^{0} = 1

= 1 \cdot \frac{6 !}{6 ! ( 6 - 6 ) !} (- 6)^{6}

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= 1 \cdot \frac{6 ! ( - 6 ) ^{6}}{6 ! ( 6 - 6 ) !}

Cancel the common factor:

6!

= 1 \cdot \frac{( - 6 ) ^{6}}{( 6 - 6 ) !}

\frac{( - 6 ) ^{6}}{( 6 - 6 ) !} = 216

\frac{( - 6 ) ^{6}}{( 6 - 6 ) !}

(- 6)^{6} = (6)^{6}

(- 6)^{6}

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 6)^{6} = (6)^{6}

= (6)^{6}

= \frac{( 6 ) ^{6}}{( 6 - 6 ) !}

(6 - 6)! = 1

(6 - 6)!

Subtract the numbers:

6 - 6 = 0

= 0!

Apply factorial rule:

0! = 1

= 1

= \frac{( 6 ) ^{6}}{1}

Apply rule

\frac{a}{1} = a

= (6)^{6}

Apply radical rule:

a = a^{\frac{1}{2}}

= (6^{\frac{1}{2}})^{6}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 6^{\frac{1}{2} \cdot 6}

\frac{1}{2} \cdot 6 = 3

\frac{1}{2} \cdot 6

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 6}{2}

Multiply the numbers:

1 \cdot 6 = 6

= \frac{6}{2}

Divide the numbers:

\frac{6}{2} = 3

= 3

= 6^{3}

6^{3} = 216

= 216

= 1 \cdot 216

Multiply the numbers:

1 \cdot 216 = 216

= 216

= 8 - 483 + 360 - 4803 + 1080 - 4323 + 216

Simplify

8 - 483 + 360 - 4803 + 1080 - 4323 + 216 : 1664 - 9603

8 - 483 + 360 - 4803 + 1080 - 4323 + 216

Add similar elements:

- 483 - 4803 - 4323 = - 9603

= 8 - 9603 + 360 + 1080 + 216

Add the numbers:

8 + 360 + 1080 + 216 = 1664

= 1664 - 9603

= 1664 - 9603

= \frac{1664 - 960 3}{4 ^{6}}

Factor

1664 - 9603 : 64 (26 - 153)

1664 - 9603

Rewrite as

= 64 \cdot 26 - 64 \cdot 153

Factor out common term

64

= 64 (26 - 153)

= \frac{64 ( 26 - 15 3 )}{4 ^{6}}

Factor

64 : 2^{6}

Factor

64 = 2^{6}

Factor

4^{6} : 2^{12}

Factor

4 = 2^{2}

= (2^{2})^{6}

Simplify

(2^{2})^{6} : 2^{12}

(2^{2})^{6}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 2^{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= 2^{12}

= 2^{12}

= \frac{2 ^{6} ( 26 - 15 3 )}{2 ^{12}}

Cancel

\frac{2 ^{6} ( 26 - 15 3 )}{2 ^{12}} : \frac{26 - 15 3}{2 ^{6}}

\frac{2 ^{6} ( 26 - 15 3 )}{2 ^{12}}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{6}}{2 ^{12}} = \frac{1}{2 ^{12 - 6}}

= \frac{26 - 15 3}{2 ^{12 - 6}}

Subtract the numbers:

12 - 6 = 6

= \frac{26 - 15 3}{2 ^{6}}

= \frac{26 - 15 3}{2 ^{6}}

2^{6} = 64

= \frac{26 - 15 3}{64}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{26 - 15 3}{64} = \frac{26}{64} - \frac{15 3}{64}

= \frac{26}{64} - \frac{15 3}{64}

Cancel

\frac{26}{64} : \frac{13}{32}

\frac{26}{64}

Cancel the common factor:

2

= \frac{13}{32}

= \frac{13}{32} - \frac{15 3}{64}

= \frac{13}{32} - \frac{15 3}{64}

= \frac{13}{32} - \frac{15 3}{64}

Manipulating right side

(\frac{1 + cos ( 21 0 ^{\circ} )}{2})^{3}

Expand

(\frac{1 + cos ( 21 0 ^{\circ} )}{2})^{3} : \frac{13}{32} - \frac{15 3}{64}

(\frac{1 + cos ( 21 0 ^{\circ} )}{2})^{3}

\frac{1 + cos ( 21 0 ^{\circ} )}{2} = \frac{2 - 3}{4}

\frac{1 + cos ( 21 0 ^{\circ} )}{2}

1 + cos (21 0^{\circ}) = 1 - \frac{3}{2}

1 + cos (21 0^{\circ})

cos (21 0^{\circ}) = - \frac{3}{2}

cos (21 0^{\circ})

Rewrite using trig identities:

cos (18 0^{\circ}) cos (3 0^{\circ}) - sin (18 0^{\circ}) sin (3 0^{\circ})

cos (21 0^{\circ})

Write

cos (21 0^{\circ})

cos (18 0^{\circ} + 3 0^{\circ})

= cos (18 0^{\circ} + 3 0^{\circ})

Use the Angle Sum identity:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= cos (18 0^{\circ}) cos (3 0^{\circ}) - sin (18 0^{\circ}) sin (3 0^{\circ})

= cos (18 0^{\circ}) cos (3 0^{\circ}) - sin (18 0^{\circ}) sin (3 0^{\circ})

Use the following trivial identity:

cos (18 0^{\circ}) = (- 1)

cos (18 0^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= (- 1)

Use the following trivial identity:

cos (3 0^{\circ}) = \frac{3}{2}

cos (3 0^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{3}{2}

Use the following trivial identity:

sin (18 0^{\circ}) = 0

sin (18 0^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 0

Use the following trivial identity:

sin (3 0^{\circ}) = \frac{1}{2}

sin (3 0^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{1}{2}

= (- 1) \frac{3}{2} - 0 \cdot \frac{1}{2}

Simplify

= - \frac{3}{2}

= 1 - \frac{3}{2}

= \frac{1 - \frac{3}{2}}{2}

Join

1 - \frac{3}{2} : \frac{2 - 3}{2}

1 - \frac{3}{2}

Convert element to fraction:

1 = \frac{1 \cdot 2}{2}

= \frac{1 \cdot 2}{2} - \frac{3}{2}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot 2 - 3}{2}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2 - 3}{2}

= \frac{\frac{2 - 3}{2}}{2}

Apply the fraction rule:

\frac{\frac{b}{c}}{a} = \frac{b}{c \cdot a}

= \frac{2 - 3}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 - 3}{4}

= (\frac{2 - 3}{4})^{3}

Simplify

(\frac{2 - 3}{4})^{3}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 2 - 3 ) ^{3}}{4 ^{3}}

(2 - 3)^{3} = 26 - 153

(2 - 3)^{3}

Apply Perfect Cube Formula:

(a - b)^{3} = a^{3} - 3 a^{2} b + 3 a b^{2} - b^{3}

a = 2, b = 3

= 2^{3} - 3 \cdot 2^{2} 3 + 3 \cdot 2 (3)^{2} - (3)^{3}

Simplify

2^{3} - 3 \cdot 2^{2} 3 + 3 \cdot 2 (3)^{2} - (3)^{3} : 26 - 153

2^{3} - 3 \cdot 2^{2} 3 + 3 \cdot 2 (3)^{2} - (3)^{3}

2^{3} = 8

2^{3}

2^{3} = 8

= 8

3 \cdot 2^{2} 3 = 123

3 \cdot 2^{2} 3

2^{2} = 4

= 3 \cdot 43

Multiply the numbers:

3 \cdot 4 = 12

= 123

3 \cdot 2 (3)^{2} = 18

3 \cdot 2 (3)^{2}

(3)^{2} = 3

(3)^{2}

Apply radical rule:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 3

= 3 \cdot 2 \cdot 3

Multiply the numbers:

3 \cdot 2 \cdot 3 = 18

= 18

(3)^{3} = 33

(3)^{3}

Apply radical rule:

a = a^{\frac{1}{2}}

= (3^{\frac{1}{2}})^{3}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= 3^{\frac{1}{2} \cdot 3}

\frac{1}{2} \cdot 3 = \frac{3}{2}

\frac{1}{2} \cdot 3

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 3}{2}

Multiply the numbers:

1 \cdot 3 = 3

= \frac{3}{2}

= 3^{\frac{3}{2}}

3^{\frac{3}{2}} = 33

3^{\frac{3}{2}}

3^{\frac{3}{2}} = 3^{1 + \frac{1}{2}}

= 3^{1 + \frac{1}{2}}

Apply exponent rule:

x^{a + b} = x^{a} x^{b}

= 3^{1} \cdot 3^{\frac{1}{2}}

Refine

= 33

= 33

= 8 - 123 + 18 - 33

Add similar elements:

- 123 - 33 = - 153

= 8 - 153 + 18

Add the numbers:

8 + 18 = 26

= 26 - 153

= 26 - 153

= \frac{26 - 15 3}{4 ^{3}}

4^{3} = 64

= \frac{26 - 15 3}{64}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{26 - 15 3}{64} = \frac{26}{64} - \frac{15 3}{64}

= \frac{26}{64} - \frac{15 3}{64}

Cancel

\frac{26}{64} : \frac{13}{32}

\frac{26}{64}

Cancel the common factor:

2

= \frac{13}{32}

= \frac{13}{32} - \frac{15 3}{64}

= \frac{13}{32} - \frac{15 3}{64}

= \frac{13}{32} - \frac{15 3}{64}

We showed that the two sides could take the same form

\Rightarrow True

Popular Examples

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prove (cos(θ)+cos(3θ))/(2cos(2θ))=cos(θ)

Frequently Asked Questions (FAQ)

Is cos^6(105)=((1+cos(210))/2)^3 ?
The answer to whether cos^6(105)=((1+cos(210))/2)^3 is True