Find an expression for cos 3x in terms of cos x?

To find an expression for cos 3x in terms of cos x, we will use trigonometric identities.

Answer: The expression for cos 3x in terms of cos x is 4 cos³x - 3 cos x.

Here's is the detailed solution.

Explanation:

As we know that cos 3x can be expressed in terms of cos (2x + x)

By using cos (A + B) = cos A × cos B - sin A × sin B

⇒ cos (2x + x) = cos 2x × cos x - sin 2x × sin x   

As we know cos 2x = 2 cos²x - 1 and sin 2x = 2 sin x cos x

⇒ (-1 + 2 cos²x) cosx - 2 sin x cos x sin x     

⇒ 2 cos³x - cos x - 2 sin²x cos x

By using sin²x = 1 - cos²x, we get

⇒ 2 cos³x - cos x - 2 (1 - cos²x) cosx   

⇒ 2 cos³x - cos x - 2 cos x  + 2cos³x

⇒ 4 cos³x - 3 cos x 

Thus, the expression for cos 3x in terms of cos x is 4cos³x - 3 cos x.