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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Apply exponent rule:

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

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

Express with sin, cos

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

Rewrite using trig identities

= 1

We showed that the two sides could take the same form

\Rightarrow True

sin (\frac{π}{3} + x) - sin (\frac{π}{3} - x) = sin (x)

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

tan (x) (sin (x) + cot (x) cos (x)) = sec (x)

sin (x) cos^{2} (x) + sin^{3} (x) = sin (x)

\frac{csc ( x ) + sec ( x )}{tan ( x ) + 1} = csc (x)

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