Question

Prove that `sin θ/(1 + cos θ) + (1 + cos θ)/sin θ = 2  "cosec"  θ`.

Answer 1

L.H.S. = `sin θ/(1 + cos θ) + (1 + cos θ)/sin θ`

= `(sin^2θ + (1 + cosθ)^2)/(sinθ(1 + cos θ))`

= `(sin^2θ + 1 + cos^2θ + 2 cos θ)/(sinθ(1 + cosθ))`

= `(2 + 2 cos θ)/(sinθ(1 + cos θ)`   ...(∵ sin² θ + cos² θ = 1)

= `(2(1 + cos θ))/(sin θ(1 + cos θ))`

= `2/sin θ`

= 2 cosec θ

L.H.S. = R.H.S.

Hence proved.