Is (1+tan^2(u))/(1-tan^2(u))= 1/(cos^2(u)-sin^2(u)) ?

The answer to whether (1+tan^2(u))/(1-tan^2(u))= 1/(cos^2(u)-sin^2(u)) is True

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prove (1+tan^2(u))/(1-tan^2(u))= 1/(cos^2(u)-sin^2(u))

prove

\frac{1 + tan ^{2} ( u )}{1 - tan ^{2} ( u )} = \frac{1}{cos ^{2} ( u ) - sin ^{2} ( u )}

True

Solution steps

\frac{1 + tan ^{2} ( u )}{1 - tan ^{2} ( u )} = \frac{1}{cos ^{2} ( u ) - sin ^{2} ( u )}

Manipulating left side

\frac{1 + tan ^{2} ( u )}{1 - tan ^{2} ( u )}

Express with sin, cos

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

Rewrite using trig identities

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

We showed that the two sides could take the same form

\Rightarrow True

csc (θ) + cot (θ) = \frac{sin ( θ )}{1 - cos ( θ )}

1 + tan^{2} (- x) = sec^{2} (x)

tan (\frac{π}{2} - x) = cot (x)

sec (θ) = \frac{tan ( θ )}{sin ( θ )}

sec (x) cos (x) = 1

Is (1+tan^2(u))/(1-tan^2(u))= 1/(cos^2(u)-sin^2(u)) ?
The answer to whether (1+tan^2(u))/(1-tan^2(u))= 1/(cos^2(u)-sin^2(u)) is True