8csc^2 theta - 3cot^2 theta = 3 + 5csc^2 theta

Question

Verify identity

Answer 1

Doing left side of identity (general practice is to operate only on one side).

(8 / sin² θ) - (3 cos² θ / sin² θ)

(8 - 3 cos² θ) / sin² θ

(8 - 3(1 - sin² θ)) / sin² θ

(8 - 3 + sin² θ) / sin² θ

(5 + 3 sin² θ) / sin² θ

(5 / sin² θ) + 3

5 csc² θ + 3

3 + 5 csc² θ

Q.E.D.

Answer 2

8csc^2 theta - 3cot^2 theta = 3 + 5csc^2 theta

8/y²-3(x/y)²=3+5/y²

8-3x²=3y²+5

3=3(x²+y²)=3

where cos(theta)=x, sin(theta)=y in unit circle