Prove tan ²φ+ cot ²φ+ 2= sec ²φcosec ²φ.

Consider the left hand side:

= tan ² φ + cot ² φ + 2
= sec ² φ - 1 + cosec ² φ - 1 + 2
[Using Identity 1 + cot ² φ = cosec ² φ and 1 + tan ² φ = sec ² φ]

= sec ² φ + cosec ² φ

= sec ² φ cosec ² φ