Solve the trigonometric equation: $\cos (3x)-\sin(x)=\sqrt 3(\cos (x)-\sin(3x))$

Question

Solve the trigonometric equation:

$$\cos (3x)-\sin(x)=\sqrt 3(\cos (x)-\sin(3x))$$

My answer is contradictory to Wolfram Alpha.

Because, W.A. gives me:

$x = \pi n - \frac {11 \pi}{12}, n \in \mathbb{ Z}$

$x = \pi n - \frac {7 \pi}{12}, n \in \mathbb{ Z}$

$x = \pi n - \frac {3 \pi}{12}, n \in \mathbb{ Z}$

But, my answer is:

$x=\frac {\pi}{12}+\pi k, k\in\mathbb{Z}$

$x=\frac {\pi}{8}+\frac {\pi k}{2}, k\in\mathbb{Z}$

Is my solution wrong? Or What is the problem in my solution?

Answer 1

$$\dfrac\pi{12}+\pi k=\pi n-\dfrac{11\pi}{12}$$

$$\iff k=n-1$$

Now for odd $k,k=2m+1$(say)

$\dfrac\pi8+\dfrac{\pi k}2=\dfrac\pi8+\dfrac{\pi(2m+1)}2=m\pi+\dfrac{5\pi}8=(m+1)\pi-\dfrac{3\pi}8$

For even $k,k=2m$(say), $\dfrac\pi8+\dfrac{\pi k}2=m\pi+\dfrac\pi8=(m+1)\pi-\dfrac{7\pi}8$

So, there must be mistake in the W.A. unless there is some typo in your input

Answer 2

$$\frac{\cos (3x)-\sin(x)}{\cos (x)-\sin(3x)}=\sqrt 3$$ $$\frac{\sin (\pi/2-3x)-\sin(x)}{\sin (\pi/2-x)-\sin(3x)}=\sqrt 3$$ $$\frac{2\cos(\pi/4-x)\sin(\pi/4-2x)}{2\cos(\pi/4+x)\sin(\pi/4-2x)}=\sqrt 3$$ $$\frac{\cos(\pi/4-x)}{\cos(\pi/4+x)}=\sqrt 3$$ $$\frac{\sin(\pi/2-\pi/4+x)}{\cos(\pi/4+x)}=\sqrt 3$$ $$\tan(\pi/4+x)=\tan(\pi/3)$$ then $$\pi/4+x=k\pi+\pi/3$$ or $$x=k\pi+\pi/12$$