Is cos^2(x)(sec^2(x)-1)=sin^2(x) ?

The answer to whether cos^2(x)(sec^2(x)-1)=sin^2(x) is True

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prove cos^2(x)(sec^2(x)-1)=sin^2(x)

prove

cos^{2} (x) (sec^{2} (x) - 1) = sin^{2} (x)

True

Solution steps

cos^{2} (x) (sec^{2} (x) - 1) = sin^{2} (x)

Manipulating left side

cos^{2} (x) (sec^{2} (x) - 1)

Express with sin, cos

(- 1 + sec^{2} (x)) cos^{2} (x)

Use the basic trigonometric identity:

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

= (- 1 + (\frac{1}{cos ( x )})^{2}) cos^{2} (x)

Simplify

(- 1 + (\frac{1}{cos ( x )})^{2}) cos^{2} (x) : - cos^{2} (x) + 1

(- 1 + (\frac{1}{cos ( x )})^{2}) cos^{2} (x)

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

(\frac{1}{cos ( x )})^{2}

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

= cos^{2} (x) (\frac{1}{cos ^{2} ( x )} - 1)

Join

- 1 + \frac{1}{cos ^{2} ( x )} : \frac{- cos ^{2} ( x ) + 1}{cos ^{2} ( x )}

- 1 + \frac{1}{cos ^{2} ( x )}

Convert element to fraction:

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

= - \frac{1 \cdot cos ^{2} ( x )}{cos ^{2} ( x )} + \frac{1}{cos ^{2} ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{- 1 \cdot cos ^{2} ( x ) + 1}{cos ^{2} ( x )}

Multiply:

1 \cdot cos^{2} (x) = cos^{2} (x)

= \frac{- cos ^{2} ( x ) + 1}{cos ^{2} ( x )}

= \frac{- cos ^{2} ( x ) + 1}{cos ^{2} ( x )} cos^{2} (x)

Multiply fractions:

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

= \frac{( - cos ^{2} ( x ) + 1 ) cos ^{2} ( x )}{cos ^{2} ( x )}

Cancel the common factor:

cos^{2} (x)

= - - cos^{2} (x) + 1

= - cos^{2} (x) + 1

= 1 - cos^{2} (x)

Rewrite using trig identities

1 - cos^{2} (x)

Use the Pythagorean identity:

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

1 - cos^{2} (x) = sin^{2} (x)

= sin^{2} (x)

= sin^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

\frac{csc ( x ) - cot ( x )}{sec ( x ) - 1} = cot (x)

\frac{1 + cos ( x )}{1 - cos ( x )} - \frac{1 - cos ( x )}{1 + cos ( x )} = 4 cot (x) csc (x)

\frac{1 - tan ^{2} ( x )}{cot ^{2} ( x ) - 1} = tan^{2} (x)

\frac{2 tan ( θ )}{1 + tan ^{2} ( θ )} = sin (2 θ)

tan^{2} (θ) sin^{2} (θ) = tan^{2} (θ) - sin^{2} (θ)

Is cos^2(x)(sec^2(x)-1)=sin^2(x) ?
The answer to whether cos^2(x)(sec^2(x)-1)=sin^2(x) is True