Prove cos 3x = 4cos3x−3cosx

We can prove cos 3x = 4cos3x−3cosx using  trigonometric identity

Answer: cos 3x = 4cos3x−3cosx

Let us see how we can prove this.

Explanation:

cos 3x can be written as  cos3x = cos(2x+x)—–(i)

As we know the trigonometric identity: cos(a+b) = cos(a)cos(b) - sin(a)sin(b)

Applying the trigonometric identity to equation (i) and we get,

cos 3x = cos2x cosx − sin2x sinx

= (−1+2cos²x)cosx − 2cosx sinx sinx

= −cosx+2cos³x − 2sin²x cosx

= −cosx+2cos ³x − 2(1−cos²x)cosx

= −3cosx + 4cos³x

=  4cos³x − 3cosx

Hence, cos 3x = 4cos3x−3cosx