Question

Verify the identity.
(1 + tan^2u)(1-sin^2u) =1

Answer 1

We have:
\[
(1+\tan^2(u))(1-\sin^2(u))=1
\]We know:
\[
1+\tan^2(u)=\sec^2(u)
\]And:
\[
\sin^2(u)+\cos^2(u)=1\iff\\
1-\sin^2(u)=\cos^2(u)
\]So:
\[
(\sec^2(u))(\cos^2(u))=\frac{\cos^2(u)}{\cos^2(u)}=1
\]