Prove the following identity:

$\Big(\dfrac{1}{1 + \tan^2 A}\Big) + \Big(\dfrac{1}{1 + \cot^2 A}\Big) = 1$

Solving L.H.S of equation,

$\Big(\dfrac{1}{1 + \tan^2 A}\Big) + \Big(\dfrac{1}{1 + \cot^2 A}\Big) = 1$

By formula:

1 + tan² A = sec² A

1 + cot² A = cosec² A

$\Rightarrow \dfrac{1}{\sec^2 A} + \dfrac{1}{\cosec^2 A} \\[1em] \Rightarrow \cos^2 A + \sin^2 A\\[1em] \text{ By formula, } \sin^2 A + \cos^2 A = 1 \\[1em] \Rightarrow 1.$

Since, L.H.S. = R.H.S.,

Hence, proved that $\Big(\dfrac{1}{1 + \tan^2 A}\Big) + \Big(\dfrac{1}{1 + \cot^2 A}\Big) = 1$.