5 cos2 𝛽 − 5 sin2 𝛽 = 5 − 10 sin2 𝛽

Question

Verify the identity. (Simplify at each step.)

Answer 1

Recall that cos²β + sin²β = 1. So, cos²β = 1 - sin²β.

Therefore, 5cos²β - 5sin²β = 5(1 - sin²β) - 5sin²β= 5 - 10sin²β

Answer 2

5cos^2B - 5sin^2B =5 -10sin^2B

divide by 5

cos^2B -sin^2B =1-2sin^2B

cos^2B + sin^2B = 1 which is a version of the Pythagorean Theorem

(a/h)^2 + (o/h)^2 = 1^2 where a=adjacent side, o=opposite side, 1 = hypotenuse on a unit circle

or you'll find a hundred+ proofs of the Pythagorean Theorem on line, including one by deceased former President James A. Garfield, a geometric proof.