Question

If $ \cos \,x\ne \frac{1}{2}, $ then the solutions of $ \cos \,x+\cos \,2x+\cos \,3x=0 $ are

• A. $ 2n\pi \pm \frac{\pi }{4},n\in Z $
• B. $ 2n\pi \pm \frac{\pi }{3},n\in Z $
• C. $ 2n\pi \pm \frac{\pi }{6},n\in Z $
• D. $ 2n\pi \pm \frac{\pi }{2},n\in Z $

Answer 1

Answer: (A)

$ \cos \,\,x+\cos \,3x\,+cos\,2x=0 $
$ \Rightarrow $ $ 2\,\cos \,2x\cos x+\cos \,2x=0 $
$ \Rightarrow $ $ \cos \,2x(2\,cos\,x+1)=0 $
$ \Rightarrow $ $ \cos \,\,2x=0\,\,\,\,\left( \because \,\,\cos \,x\ne -\frac{1}{2} \right) $
$ \Rightarrow $ $ 2x=\frac{\pi }{2} $
$ \Rightarrow $ $ x=\frac{\pi }{4} $
$ \therefore $ General value is $ 2n\pi \pm \frac{\pi }{4},\,\,n\in Z $