Question

Prove that 1+ cos theta - sin^2theta/
sin theta(1 + cos theta)
= cot theta

Answer 1

To prove :-

(1 + cos∅ - sin²∅)/[sin∅(1 + cos∅)] = cot∅

LHS :-

= (1 + cos∅ - sin²∅)/[sin∅(1 + cos∅)]

= (cos∅ + 1 - sin²∅)/[sin∅(1 + cos∅)]

using identity sin²∅ + cos∅ = 1 ➡ cos²∅ = 1 - sin²∅

= (cos∅ + cos²∅)/[sin∅(1 + cos∅)]

taking cos∅ as common in the numerator

= [cos∅(1 + cos∅)]/[sin∅(1 + cos∅)]

by canceling (1 + cos∅), we get

= cos∅/sin∅

= cot∅ (since cot∅ = cos∅/sin∅)

➡ cot∅ = cot∅

LHS = RHS. hence proved!