Is sec(t)(csc(t)(tan(t)+cot(t)))=sec^2(t)+csc^2(t) ?

The answer to whether sec(t)(csc(t)(tan(t)+cot(t)))=sec^2(t)+csc^2(t) is True

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Popular Trigonometry >

prove sec(t)(csc(t)(tan(t)+cot(t)))=sec^2(t)+csc^2(t)

Solution

prove

sec (t) (csc (t) (tan (t) + cot (t))) = sec^{2} (t) + csc^{2} (t)

Solution

True

Solution steps

sec (t) csc (t) (tan (t) + cot (t)) = sec^{2} (t) + csc^{2} (t)

Manipulating left side

sec (t) csc (t) (tan (t) + cot (t))

Express with sin, cos

(cot (t) + tan (t)) csc (t) sec (t)

Use the basic trigonometric identity:

cot (x) = \frac{cos ( x )}{sin ( x )}

= (\frac{cos ( t )}{sin ( t )} + tan (t)) csc (t) sec (t)

Use the basic trigonometric identity:

tan (x) = \frac{sin ( x )}{cos ( x )}

= (\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}) csc (t) sec (t)

Use the basic trigonometric identity:

csc (x) = \frac{1}{sin ( x )}

= (\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}) \frac{1}{sin ( t )} sec (t)

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

= (\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}) \frac{1}{sin ( t )} \cdot \frac{1}{cos ( t )}

Simplify

(\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}) \frac{1}{sin ( t )} \cdot \frac{1}{cos ( t )} : \frac{cos ^{2} ( t ) + sin ^{2} ( t )}{sin ^{2} ( t ) cos ^{2} ( t )}

(\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}) \frac{1}{sin ( t )} \cdot \frac{1}{cos ( t )}

Multiply fractions:

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

= \frac{1 \cdot 1 \cdot ( \frac{c o s ( t )}{s i n ( t )} + \frac{s i n ( t )}{c o s ( t )} )}{sin ( t ) cos ( t )}

1 \cdot 1 \cdot (\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}) = \frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}

1 \cdot 1 \cdot (\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )})

Multiply:

1 \cdot 1 \cdot (\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}) = (\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )})

= (\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )})

Remove parentheses:

(a) = a

= \frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}

= \frac{\frac{c o s ( t )}{s i n ( t )} + \frac{s i n ( t )}{c o s ( t )}}{sin ( t ) cos ( t )}

Join

\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )} : \frac{cos ^{2} ( t ) + sin ^{2} ( t )}{sin ( t ) cos ( t )}

\frac{cos ( t )}{sin ( t )} + \frac{sin ( t )}{cos ( t )}

Least Common Multiplier of

sin (t), cos (t) : sin (t) cos (t)

sin (t), cos (t)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sin (t)

cos (t)

= sin (t) cos (t)

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

sin (t) cos (t)

For

\frac{cos ( t )}{sin ( t )} :

multiply the denominator and numerator by

cos (t)

\frac{cos ( t )}{sin ( t )} = \frac{cos ( t ) cos ( t )}{sin ( t ) cos ( t )} = \frac{cos ^{2} ( t )}{sin ( t ) cos ( t )}

For

\frac{sin ( t )}{cos ( t )} :

multiply the denominator and numerator by

sin (t)

\frac{sin ( t )}{cos ( t )} = \frac{sin ( t ) sin ( t )}{cos ( t ) sin ( t )} = \frac{sin ^{2} ( t )}{sin ( t ) cos ( t )}

= \frac{cos ^{2} ( t )}{sin ( t ) cos ( t )} + \frac{sin ^{2} ( t )}{sin ( t ) cos ( t )}

Since the denominators are equal, combine the fractions:

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

= \frac{cos ^{2} ( t ) + sin ^{2} ( t )}{sin ( t ) cos ( t )}

= \frac{\frac{c o s ^{2} ( t ) + s i n ^{2} ( t )}{s i n ( t ) c o s ( t )}}{sin ( t ) cos ( t )}

Apply the fraction rule:

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

= \frac{cos ^{2} ( t ) + sin ^{2} ( t )}{sin ( t ) cos ( t ) sin ( t ) cos ( t )}

sin (t) cos (t) sin (t) cos (t) = sin^{2} (t) cos^{2} (t)

sin (t) cos (t) sin (t) cos (t)

Apply exponent rule:

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

cos (t) cos (t) = cos^{1 + 1} (t)

= sin (t) sin (t) cos^{1 + 1} (t)

Add the numbers:

1 + 1 = 2

= sin (t) sin (t) cos^{2} (t)

Apply exponent rule:

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

sin (t) sin (t) = sin^{1 + 1} (t)

= sin^{1 + 1} (t) cos^{2} (t)

Add the numbers:

1 + 1 = 2

= sin^{2} (t) cos^{2} (t)

= \frac{cos ^{2} ( t ) + sin ^{2} ( t )}{sin ^{2} ( t ) cos ^{2} ( t )}

= \frac{cos ^{2} ( t ) + sin ^{2} ( t )}{sin ^{2} ( t ) cos ^{2} ( t )}

= \frac{cos ^{2} ( t ) + sin ^{2} ( t )}{cos ^{2} ( t ) sin ^{2} ( t )}

Rewrite using trig identities

Use the basic trigonometric identity:

sin (x) = \frac{1}{csc ( x )}

\frac{cos ^{2} ( t ) + ( \frac{1}{c s c ( t )} ) ^{2}}{cos ^{2} ( t ) ( \frac{1}{c s c ( t )} ) ^{2}}

Use the basic trigonometric identity:

cos (x) = \frac{1}{sec ( x )}

\frac{( \frac{1}{s e c ( t )} ) ^{2} + ( \frac{1}{c s c ( t )} ) ^{2}}{( \frac{1}{s e c ( t )} ) ^{2} ( \frac{1}{c s c ( t )} ) ^{2}}

Simplify

\frac{( \frac{1}{s e c ( t )} ) ^{2} + ( \frac{1}{c s c ( t )} ) ^{2}}{( \frac{1}{s e c ( t )} ) ^{2} ( \frac{1}{c s c ( t )} ) ^{2}}

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

(\frac{1}{sec ( t )})^{2}

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

= \frac{( \frac{1}{s e c ( t )} ) ^{2} + ( \frac{1}{c s c ( t )} ) ^{2}}{( \frac{1}{c s c ( t )} ) ^{2} \frac{1}{s e c ^{2} ( t )}}

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

(\frac{1}{csc ( t )})^{2}

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

= \frac{( \frac{1}{s e c ( t )} ) ^{2} + ( \frac{1}{c s c ( t )} ) ^{2}}{\frac{1}{s e c ^{2} ( t )} \cdot \frac{1}{c s c ^{2} ( t )}}

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

(\frac{1}{sec ( t )})^{2}

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

(\frac{1}{csc ( t )})^{2}

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

= \frac{\frac{1}{s e c ^{2} ( t )} + \frac{1}{c s c ^{2} ( t )}}{\frac{1}{s e c ^{2} ( t )} \cdot \frac{1}{c s c ^{2} ( t )}}

Multiply

\frac{1}{sec ^{2} ( t )} \cdot \frac{1}{csc ^{2} ( t )} : \frac{1}{sec ^{2} ( t ) csc ^{2} ( t )}

\frac{1}{sec ^{2} ( t )} \cdot \frac{1}{csc ^{2} ( t )}

Multiply fractions:

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

= \frac{1 \cdot 1}{sec ^{2} ( t ) csc ^{2} ( t )}

Multiply the numbers:

1 \cdot 1 = 1

= \frac{1}{sec ^{2} ( t ) csc ^{2} ( t )}

= \frac{\frac{1}{s e c ^{2} ( t )} + \frac{1}{c s c ^{2} ( t )}}{\frac{1}{s e c ^{2} ( t ) c s c ^{2} ( t )}}

Join

\frac{1}{sec ^{2} ( t )} + \frac{1}{csc ^{2} ( t )} : \frac{csc ^{2} ( t ) + sec ^{2} ( t )}{sec ^{2} ( t ) csc ^{2} ( t )}

\frac{1}{sec ^{2} ( t )} + \frac{1}{csc ^{2} ( t )}

Least Common Multiplier of

sec^{2} (t), csc^{2} (t) : sec^{2} (t) csc^{2} (t)

sec^{2} (t), csc^{2} (t)

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

sec^{2} (t)

csc^{2} (t)

= sec^{2} (t) csc^{2} (t)

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

sec^{2} (t) csc^{2} (t)

For

\frac{1}{sec ^{2} ( t )} :

multiply the denominator and numerator by

csc^{2} (t)

\frac{1}{sec ^{2} ( t )} = \frac{1 \cdot csc ^{2} ( t )}{sec ^{2} ( t ) csc ^{2} ( t )} = \frac{csc ^{2} ( t )}{sec ^{2} ( t ) csc ^{2} ( t )}

For

\frac{1}{csc ^{2} ( t )} :

multiply the denominator and numerator by

sec^{2} (t)

\frac{1}{csc ^{2} ( t )} = \frac{1 \cdot sec ^{2} ( t )}{csc ^{2} ( t ) sec ^{2} ( t )} = \frac{sec ^{2} ( t )}{sec ^{2} ( t ) csc ^{2} ( t )}

= \frac{csc ^{2} ( t )}{sec ^{2} ( t ) csc ^{2} ( t )} + \frac{sec ^{2} ( t )}{sec ^{2} ( t ) csc ^{2} ( t )}

Since the denominators are equal, combine the fractions:

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

= \frac{csc ^{2} ( t ) + sec ^{2} ( t )}{sec ^{2} ( t ) csc ^{2} ( t )}

= \frac{\frac{c s c ^{2} ( t ) + s e c ^{2} ( t )}{s e c ^{2} ( t ) c s c ^{2} ( t )}}{\frac{1}{s e c ^{2} ( t ) c s c ^{2} ( t )}}

Divide fractions:

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

= \frac{( csc ^{2} ( t ) + sec ^{2} ( t ) ) sec ^{2} ( t ) csc ^{2} ( t )}{sec ^{2} ( t ) csc ^{2} ( t ) \cdot 1}

Refine

= \frac{( csc ^{2} ( t ) + sec ^{2} ( t ) ) sec ^{2} ( t ) csc ^{2} ( t )}{sec ^{2} ( t ) csc ^{2} ( t )}

Cancel the common factor:

sec^{2} (t)

= \frac{( csc ^{2} ( t ) + sec ^{2} ( t ) ) csc ^{2} ( t )}{csc ^{2} ( t )}

Cancel the common factor:

csc^{2} (t)

= csc^{2} (t) + sec^{2} (t)

csc^{2} (t) + sec^{2} (t)

csc^{2} (t) + sec^{2} (t)

= sec^{2} (t) + csc^{2} (t)

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is sec(t)(csc(t)(tan(t)+cot(t)))=sec^2(t)+csc^2(t) ?
The answer to whether sec(t)(csc(t)(tan(t)+cot(t)))=sec^2(t)+csc^2(t) is True