Prove the following trigonometric identities.
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,

$\frac{1}{1+\sin A}+\frac{1}{1-\sin A}=\frac{(1-\sin A)+(1+\sin A)}{(1+\sin A)(1-\sin A)}$

$=\frac{1-\sin A+1+\sin A}{1-\sin ^{2} A}$

$=\frac{2}{\cos ^{2} A}$

$=2 \sec ^{2} A$

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