Prove the following trigonometric identity:

\[ \cos^2(2x) - \cos^2(6x) = \sin(4x) \sin(8x) \]

Cos² 2x = (1+cos 4x )/2 cos² 6x = (1+cos 12x)/2 subtract to get 1/2 (cos 4x - cos 12x ) = 1/2 (2 sin 4x sin 8x ) as cos A - cos B = 2 [ sin (A+B)/2 sin (A-B)/2 ] so the answer

Answered by Everest2017 | 2024-06-10

We have proved the identity cos 2 ( 2 x ) − cos 2 ( 6 x ) = sin ( 4 x ) sin ( 8 x ) by using the difference of squares along with cosine subtraction and addition identities. The left-hand side can be expressed in terms of sine functions, ultimately confirming the equality. This demonstrates the equivalence of both sides of the identity.
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Answered by Anonymous | 2024-09-26