Question:
Prove the following trigonometric identities.
$\frac{1}{1+\sin A}+\frac{1}{1-\sin A}=2 \sec ^{2} A$
Solution:
We have to prove $\frac{1}{1+\sin A}+\frac{1}{1-\sin A}=2 \sec ^{2} A$
We know that, $\sin ^{2} A+\cos ^{2} A=1$
So,
$\frac{1}{1+\sin A}+\frac{1}{1-\sin A}=\frac{(1-\sin A)+(1+\sin A)}{(1+\sin A)(1-\sin A)}$
$=\frac{1-\sin A+1+\sin A}{1-\sin ^{2} A}$
$=\frac{2}{\cos ^{2} A}$
$=2 \sec ^{2} A$