Prove that each equation is an identity.

Question

1) sin^4 w = (1 - cot^2 w + cos^2 w cot^2 w) / csc ^2 w

2) 1/ (cscθ - cot θ ) = (1 + cot θ) / sin θ

Answer 1

2) Assume the identity 1/(cscθ - cotθ) = (1 + cosθ)/sinθ

Left hand side identity = 1/(cscθ - cotθ)

= (cscθ + cotθ)/(cscθ - cotθ)(cscθ + cotθ)

= (cscθ + cotθ)/(csc²θ - cot²θ)

= (cscθ + cotθ)/1

= cscθ + cotθ

= 1/sinθ + cosθ/sinθ

= ( 1 + cosθ)/sinθ

Right hand side identity.

Comments

Right hand side is (1 + cot θ) / sin θ, with cotangent not cosine. 

---

The solution we had provided is correct with ( 1 + cosθ)/sinθ on right hand side .

There is no possibility  for cotangent on the right hand side .