Question:
Prove the following trigonometric identities.
$\cos ^{2} A+\frac{1}{1+\cot ^{2} A}=1$
Solution:
We know that,
$\sin ^{2} A+\cos ^{2} A=1$
$\operatorname{cosec}^{2} A-\cot ^{2} A=1$
So,
$\cos ^{2} A+\frac{1}{1+\cot ^{2} A}=\cos ^{2} A+\frac{1}{\operatorname{cosec}^{2} A}$
$=\cos ^{2} A+\left(\frac{1}{\operatorname{cosec} A}\right)^{2}$
$=\cos ^{2} A+(\sin A)^{2}$
$=\cos ^{2} A+\sin ^{2} A$
$=1$