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Find the general solution of the following equation 

Q (6)  \small \cos 3x + \cos x -\cos 2x = 0

Answers (1)

best_answer

We know that 
\cos A + \cos B = 2\cos\frac{A+B}{2}\cos\frac{A-B}{2} \\ and \\ \cos A - \cos B = -2\sin\frac{A+B}{2}\sin\frac{A-B}{2}
We use these identities
(cos3x + cosx) - cos2x = 2cos2xcosx -cos2x = 0
                                     = cos2x(2cosx-1) = 0
So, either 
 cos2x = 0                        or                                 cosx=\frac{1}{2}
        2x=(2n+1)\frac{\pi}{2}                                         cosx =\cos\frac{\pi}{3}
           x=(2n+1)\frac{\pi}{4}                                              x =2n\pi \pm \frac{\pi}{3}

\therefore the general solution is 

 x=(2n+1)\frac{\pi}{4}   \ or \ 2n\pi \pm \frac{\pi}{3}

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manish

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