Question:
$\frac{1+\tan ^{2} A}{1+\cot ^{2} A}=?$
(a) –1
(b) sec2A
(c) tan2A
(d) cot2A
Solution:
$\frac{1+\tan ^{2} A}{1+\cot ^{2} A}$
$=\frac{\sec ^{2} A}{\operatorname{cosec}^{2} A}$
$=\frac{\sin ^{2} A}{\cos ^{2} A} \quad\left(\sec \theta=\frac{1}{\cos \theta}\right.$ and $\left.\operatorname{cosec} \theta=\frac{1}{\sin \theta}\right)$
$=\tan ^{2} A$
Hence, the correct answer is option (c).