Question

Prove the identity cos(3x)=cos^3(x)-3sin^2(x)cos(x), step by step.

Answer 1

The trigonometric identity cos(3x) = cos^3(x) - 3sin^2(x)cos(x) is proven by expanding cos(3x) using angle addition formulas and trigonometric identities to arrive at the desired expression.

Step-by-step explanation:

To prove the trigonometric identity cos(3x) = cos^3(x) - 3sin^2(x)cos(x), we can use trigonometric identities and expand cos(3x) using the angle addition formula for cosine.

Starting from the angle addition formula:
cos(3x) = cos(2x + x)

Now expand using the angle addition identity cos(a + b) = cos(a)cos(b) - sin(a)sin(b):

cos(3x) = cos(2x)cos(x) - sin(2x)sin(x)

Expanding cos(2x) and sin(2x) using the double-angle formulas:

cos(2x) = cos^2(x) - sin^2(x)
sin(2x) = 2sin(x)cos(x)

Substitute back into the equation:

cos(3x) = (cos^2(x) - sin^2(x))cos(x) - (2sin(x)cos(x))sin(x)

Simplify the equation:

cos(3x) = cos^3(x) - sin^2(x)cos(x) - 2sin^2(x)cos(x)

Combine like terms:

cos(3x) = cos^3(x) - 3sin^2(x)cos(x)

Thus, we have shown that the identity holds true.

Answer 2

cos(A+B) = cosAcosB - sinAsinB
cos(3x) = cos(x+2x)
cos(x+2x)= cosxcos2x-sinxsin2x
cos2x = 2cos^2 (x) - 1
sin(2x) = 2sin(x)cos(x)
by putting the values we get
cos(x+2x)= cosx*cos2x-sinx*sin2x=cosx*(2cos^2 (x) - 1) -(sinx*2sin(x)cos(x))
=cos^3(x)-3sin^2(x)cos(x)
hope it helps