Choosing right argument expansion for trigonometric equation

Question

Given equation $$\cos(x) + \cos(2x) + \cos(3x) + \cos(4x) = 0,$$ which argument expansion will be the most convenient?

Answer 1

HINT:

Use Prosthaphaeresis Formula,

$$\cos x+\cos4x=2\cos\dfrac{5x}2\cos\dfrac{3x}2$$

$$\cos2x+\cos3x=2\cos\dfrac{5x}2\cos\dfrac x2$$

Again, $$\cos\dfrac{3x}2+\cos\dfrac x2=?$$

Now if $\cos y=0,y=(2n+1)\dfrac\pi2$ where $n$ is any integer

Answer 2

As $\sin\dfrac x2\ne0\iff x\ne2n\pi$(why?)$\ \ \ \ (1)$

using $\sum \cos$ when angles are in arithmetic progression,

$$\sin\left(4x+\dfrac x2\right)=\sin\left(x-\dfrac x2\right)$$

$$\implies\dfrac{9x}2=m\pi+(-1)^m\dfrac x2$$ where $m$ is any integer

If $m$ is even $=2r$(say), $$\dfrac{9x}2=2r\pi+\dfrac x2\iff x=\dfrac{r\pi}2$$ $(1)$ demands $4\nmid r$

If $m$ is odd, $=2r+1$(say), $$\dfrac{9x}2=(2r+1)\pi-\dfrac x2\iff x=\dfrac{(2r+1)\pi}5$$