- #1

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## Homework Statement

[tex]\cos x-\sqrt{3}\sin x=\cos(3x)[/tex]

## Homework Equations

## The Attempt at a Solution

Dividing both the sides by two i.e

[tex]\cos x \cos \frac{\pi}{3}-\sin x \sin \frac{\pi}{3}=\cos (3x)/2[/tex]

LHS can be written as ##\cos(x+\pi/3)##. Substituting ##x+\pi/3=t \Rightarrow 3x=3t-\pi \Rightarrow \cos(3x)=-\cos(3t)##

[tex]\cos t=-\cos(3t)/2 \Rightarrow 2\cos t+\cos(3t)=0[/tex]

##\because \cos(3t)=4\cos^3t-3\cos t##

[tex]4\cos^3t-\cos t=0 \Rightarrow \cos t(4\cos^2t-1)=0[/tex]

Hence, ##\cos t=0 \Rightarrow t=(2n+1)\pi/2 \Rightarrow x=(2n+1)\pi/2-\pi/3## and ##\cos t=±1/2 \Rightarrow t=k\pi ± \pi/3 \Rightarrow x=k\pi, k\pi-2\pi/3## where ##n,k \in Z##.

But these are wrong. The correct answers are: ##n\pi, \pi(3k+(-1)^k)/6##.

Any help is appreciated. Thanks!