Question

Prove that Cos³A.cos3A + sin³A.sin3A =cos³2A​

Answer 1

Answer:

You need the following identities:

1. cos^2(X) + sin^2(X) = 1

2. sin(X+Y) = sin X cosY + sin Y cos X

3. cos(X-Y) = cos X cos Y + sin X sin Y

4. sin(2X) = 2*sin X cos X

cos^3(A) cos 3A + sin^3(A) sin 3A =

cos^2(A) (cos A cos 3A) + sin^2(A)(sin A sin 3A) =

Identity 1 gives cos^2(A) = 1 - sin^2(A) and sin^2(A) = 1 - cos^2(A)

(1-sin^2(A)) (cos A cos 3A) + (1-cos^2(A))(sin A sin 3A) =

cos A cos 3A + sin A sin 3A - sin^2(A) cos A cos 3A - cos^2(A) sin A sin 3A =

cos A cos 3A + sin A sin 3A - sin A cos A(sin A cos 3A + sin 3A cosA) =

Identity 2 gives sin A cos 3A + sin 3A cos A = sin 4A

Identity 3 gives cos A cos 3A + sin A sin 3A = cos 2A

cos 2A - sin A cos A (sin 4A) =

Identity 4 gives sin 4A = 2*sin 2A cos 2A and 2*sin A cos A = sin 2A

cos 2A - 2*sin A cos A (sin 2A cos 2A) =

cos 2A - sin 2A sin 2A cos 2A =

cos 2A (1 - sin^2(2A)) =

Using Identity 1 again

cos 2A (cos^2(2A)) =

cos^3(2A)

Answer 2

Answer:

hey mate , answer is in attachment, please mark as brainliest