cos3x = 1- 3cosx Solve for x

Question

what I did:

cos(2x + x) = 1-3cosx

used compound angle formula

cos2xcosx - sin2xsinx = 1- 3cosx

used double angle formulae for both cos2x and sin2x

(1-2sin^2x)cosx - 2sin^2xcosx = 1-3cosx

but I'm stuck now.

Any help appreciated.

Answer 1

working from where you left off...

(1-2sin^2x)cosx-2sin^2xcosx+3cosx=1

factor out cosx

cosx(1-2sin^2x-2sin^2x+3)=1

cosx(4-4sin^2x)=1

4cosx(1-sin^2x)=1

4cosx(cos^2x)=1

4cos^3x=1

cos^3x=1/4

take the cube root of both sides

cosx=0.62996

use arccos finder on your calculator

x=50.95282°

this is correct because I plugged it into the original equation and got a true sentence

you can find the other angles that will also work