Question

Prove that: `(1 + "cosec"  θ)/("cosec"  θ) = (cos^2 θ)/(1 - sin θ)`

Answer 1

Given: 

Prove: `(1 + "cosec"  θ)/("cosec"  θ) = (cos^2 θ)/(1 - sin θ)`

LHS:

`(1 + "cosec"  θ)/("cosec"  θ) = 1/("cosec"  θ) + ("cosec"  θ)/("cosec"  θ)`

= sin θ + 1

= 1 + sin θ

RHS:

`(cos 2θ)/(1 - sin θ) = (1 - 2sin^2 θ)/(1 - sin θ)`

= `((1 - sin θ)(1 + sin θ))/(1 - sin θ)`

= 1 + sin θ

Therefore, LHS = 1 + sin θ and RHS = 1 + sin θ.

Hence proved, for θ where cosec θ and 1 – sin θ are defined (i.e. sin θ ≠ 0, 1).