What is the value of sec^3(pi/4)+tan^2(pi/4)sec(pi/4) ?

The value of sec^3(pi/4)+tan^2(pi/4)sec(pi/4) is 3sqrt(2)

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sec^3(pi/4)+tan^2(pi/4)sec(pi/4)

sec^{3} (\frac{π}{4}) + tan^{2} (\frac{π}{4}) sec (\frac{π}{4})

32

Decimal

4.24264 \dots

Solution steps

sec^{3} (\frac{π}{4}) + tan^{2} (\frac{π}{4}) sec (\frac{π}{4})

Use the following trivial identity:

sec (\frac{π}{4}) = 2

sec (\frac{π}{4})

sec (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

= 2

Use the following trivial identity:

tan (\frac{π}{4}) = 1

tan (\frac{π}{4})

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

= 1

= (2)^{3} + 1^{2} 2

Simplify

(2)^{3} + 1^{2} 2 : 32

(2)^{3} + 1^{2} 2

Apply rule

1^{a} = 1

1^{2} = 1

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

(2)^{3} = 22

(2)^{3}

Apply radical rule:

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

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

Apply exponent rule:

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

= 2^{\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}

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

2^{\frac{3}{2}} = 22

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

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

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

Apply exponent rule:

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

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

Refine

= 22

= 22

1 \cdot 2 = 2

1 \cdot 2

Multiply:

1 \cdot 2 = 2

= 2

= 22 + 2

Add similar elements:

22 + 2 = 32

= 32

= 32

What is the value of sec^3(pi/4)+tan^2(pi/4)sec(pi/4) ?
The value of sec^3(pi/4)+tan^2(pi/4)sec(pi/4) is 3sqrt(2)