Is sec^2(x)-cot^2(pi/2-x)=1 ?

The answer to whether sec^2(x)-cot^2(pi/2-x)=1 is True

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prove sec^2(x)-cot^2(pi/2-x)=1

prove

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

True

Solution steps

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

Manipulating left side

sec^{2} (x) - cot^{2} (\frac{π}{2} - x)

Rewrite using trig identities

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

Apply exponent rule:

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

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

Express with sin, cos

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

Rewrite using trig identities

= 1

We showed that the two sides could take the same form

\Rightarrow True

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

sin (2 x) = - 2 sin (- x) cos (- x)

sin (θ) csc (θ) - sin^{2} (θ) = cos^{2} (θ)

tan (a) cot (a) = cos^{2} (a) + sin^{2} (a)

1 + tan^{2} (t) = \frac{1}{cos ^{2} ( t )}

Is sec^2(x)-cot^2(pi/2-x)=1 ?
The answer to whether sec^2(x)-cot^2(pi/2-x)=1 is True