Prove the following identity:

tan² A + cot² A + 2 = sec² A cosec² A

Solving L.H.S of equation,

$\Rightarrow \dfrac{\sin^2 A}{\cos^2 A} + \dfrac{\cos^2 A}{\sin^2 A} + 2 \\[1em] \Rightarrow \dfrac{\sin^4 A + \cos^4 A + 2\cos^2 A \sin^2 A}{\cos^2 A \sin^2 A} \\[1em] \Rightarrow \dfrac{(\sin^2 A + \cos^2 A)^2}{\cos^2 A \sin^2 A} \\[1em] \Rightarrow \dfrac{1^2}{\cos^2 A \sin^2 A} \\[1em] \Rightarrow \dfrac{1}{\sin^2 A} \times \dfrac{1}{\cos^2 A} \\[1em] \Rightarrow \cosec^2 A \sec^2 A .$

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

Hence, proved that tan² A + cot² A + 2 = sec² A cosec² A.