(3 cos theta - 4 sin theta)^2 + (4 cos theta + 3 sin theta)^2 = 25 Verify the identity?

Question

Prove the identity using other trigonometric identities and substitutions. A calculator may not be used as a proof

Answer 1

9 cos ² θ - 24 cosθ sinθ + 16 sin ² θ
16 cos² θ + 24 cosθ sinθ + 9 sin² θ--------ADD

25 cos ² θ + 25 sin ² θ
25 ( cos ² θ + sin ² θ )
25 (as required)

Answer 2

( 3 cosθ - 4 sinθ )² + ( 4 cosθ + 3 sinθ )² =
= 9 cos²θ - 24 sinθcosθ + 16 sin²θ + 16 cos²θ + 24 sinθcosθ + 9 sin²θ = 9 (cos²θ + sin²θ) + 16 (cos²θ + sin²θ) = 9 + 16 = 25

Answer 3

let theta=x so i don't have to type "theta" a lot
then
(3cos(x)-4sin(x))^2 + (4cos(x)+3sin(x))^2=25

Step 1: Foil.
9cos^2(x) - 24sin(x)cos(x) + 16sin^2(x) + 16cos^2(x) + 24sin(x)cos(x) + 9sin^2(x) = 25

Step 2: -24sinxcosx and +24sinxcosx cancel out.
9cos^2(x) + 9sin^2(x) + 16sin^2(x) + 16cos^2(x) = 25

Step 3: Factor out constants.
9(cos^2(x) + sin^2(x)) + 16(sin^2(x) + cos^2(x)) = 25

Step 4: sin^2(x) + cos^2(x) = 1 by trig identity.
9 + 16 = 25

Step 5: Add 9 and 16
25=25

Verified!