Question

verify the identity:
cos3x=4cos^3 x-3cosx

Answer 1

\[\cos3x=4\cos ^{3}x-3cosx\]

Answer 2

well start with

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

use an appropriate substitution for cos(2x) ans sin(2x) = 2sin(x)cos(x)

try cos(2x) = 2cos^2(x) - 1

sp you have

\[ ( 2\cos^2 (x) -1)\cos(x) - (2\sin(x)\cos(x))\sin(x)\]

which simplifies to

\[2\cos^3x - \cos(x) - 2\sin^2(x)\cos(x)\]

now use the substitution sin^2(x) = 1 - cos^2(x)

so you get

\[2\cos^3(x) - \cos(x) - 2(1 - \cos^2(x))\cos(x)\]

just distribute the term on the right and collect like terms for the answer.