Question

Cos x + cos 7x =cos 4x

Answer 1

Given: $cosx+cos7x=cos4x$

To find: the solution

Solution:

• Now, $cosx+cos7x=cos4x$

• $\Rightarrow 2\:cos\frac{x+7x}{2}\:cos\frac{x-7x}{2}=cos4x$

• $\Rightarrow 2\:cos4x\:cos3x=cos4x$

• $\Rightarrow 2\:cos4x\:cos3x-cos4x=0$

• $\Rightarrow cos4x\:(2\:cos3x-1)=0$

• $\Rightarrow cos4x=0\:or,\:2\:cos3x=1$

• $\Rightarrow cos4x=cos\frac{\pi}{2}\:or,\:cos3x=\frac{1}{2}=cos\frac{\pi}{3}$

• $\Rightarrow 4x=\frac{\pi}{2}\:or,\:3x=\frac{\pi}{3}$

• $\Rightarrow x=\frac{\pi}{8}\:or,\:\frac{\pi}{9}$

Answer: required solution is $x=\frac{\pi}{8},\:\frac{\pi}{9}$

To remember:

• $cosC+cosD=2\:cos\frac{C+D}{2}\:cos\frac{C-D}{2}$
• $cosC-cosD=2\:sin\frac{C+D}{2}\:sin\frac{D-C}{2}$