Question

Determine whether the equation is an identity.

cos(3 theta) = 4(cos(theta))^3 - 3 cos(theta)

Answer 1

\[\cos(3 \theta) =\cos(2 \theta+\theta)=\cos(2 \theta)\cos(\theta)-\sin(2 \theta)\sin(\theta)\]

Answer 2

\[you know that \cos(2 \theta)=\cos^2(\theta)-\sin^2(\theta)\]

Answer 3

\[\cos(2 \theta + \theta)=(\cos^2(\theta)-\sin^2(\theta))\cos(\theta)-2\sin(\theta)\cos(\theta)\sin(\theta)\]

Answer 4

\[i used other trig identity: \sin(2 \theta)=2\sin(\theta)\cos(\theta)\]

Answer 5

further:\[\cos^3(\theta)-(1-\cos^2(\theta))\cos(\theta)-2(1-\cos^2(\theta))\cos(\theta)\]

Answer 6

I used other trig identity:\[\sin^2(\theta)=1-\cos^2(\theta)\]

Answer 7

so finally opening all brakets we have:\[\cos^3(\theta)-\cos(\theta)+\cos^3(\theta)-2\cos(\theta)+2\cos^3(\theta)=4\cos^3(\theta)-3\cos(\theta)\]

Answer 8

good luck