Question:
Prove the following trigonometric identities.
if $\cos A+\cos ^{2} A=1$, prove that $\sin ^{2} A+\sin ^{4} A=1$
Solution:
Given: $\cos A+\cos ^{2} A=1$
We have to prove $\sin ^{2} A+\sin ^{4} A=1$
Now,
$\cos A+\cos ^{2} A=1$
$\Rightarrow \quad \cos A=1-\cos ^{2} A$
$\Rightarrow \quad \cos A=\sin ^{2} A$
$\Rightarrow \quad \sin ^{2} A=\cos A$
Therefore, we have
$\sin ^{2} A+\sin ^{4} A=\cos A+(\cos A)^{2}$
$=\cos A+\cos ^{2} A$
$=1$
Hence proved.