Question

Prove the following:

(sinθ + cosecθ)² + (cosθ + secθ)² = tan²θ + cot²θ + 7

Answer 1

L.H.S. = (sinθ + cosecθ)² + (cosθ + secθ)²  

= sin²θ + cosec²θ + 2sinθ cosecθ + cos²θ + sec²θ + 2cosθ secθ

= (sin²θ + cos²θ) + cosec²θ + 2 + sec²θ + 2

= 1 + (1 + cot²θ) + 2 + (1 + tan²θ) + 2

= tan²θ + cot²θ + 7

= R.H.S.