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

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

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

prove

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

True

Solution steps

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

Manipulating left side

sec^{2} (\frac{π}{2} - x) - 1

Rewrite using trig identities

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

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

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

Manipulating right side

cot^{2} (x)

Express with sin, cos

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

Rewrite using trig identities

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

Expand

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

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

We showed that the two sides could take the same form

\Rightarrow True

4 cos^{2} (x) + 4 sin^{2} (x) = 4

tan (θ) + cot (θ) = (sec (θ)) (csc (θ))

csc (w) - sin (w) = cot (w) cos (w)

\frac{csc ( t ) + 1}{cot ( t )} = \frac{cot ( t )}{csc ( t ) - 1}

\frac{cos ( x )}{sin ^{2} ( x )} = csc (x) cot (x)

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