Question

Prove the following:

`1 + (cot^2 alpha)/(1 + "cosec"  alpha)` = cosec α

Answer 1

L.H.S = `1 + (cot^2 alpha)/(1 + "cosec"  alpha)`

= `1 + ((cos^2 alpha)/(sin^2 alpha))/((1 + 1)/(sin alpha))`  ...`[∵ cot theta = (cos theta)/(sin theta) "and"  "cosec"  theta = 1/sin theta]`

= `1 + (cos^2 alpha)/(sinalpha (1 + sin alpha))`

= `(sin alpha(1 + sin alpha) + cos^2 alpha)/(sin alpha(1 + sin alpha))`

= `(sin alpha + (sin^2 alpha + cos^2 alpha))/(sin alpha(1 + sin alpha)`  ...[∵ sin²θ + cos²θ = 1]

= `((sin alpha + 1))/(sin alpha(sin alpha + 1))` 

= `1/sinalpha` ...`[∵ "cosec"  theta = 1/sin theta]`

= cosec α 

= R.H.S