Question

In sin x + sin 2 x = 1 , then cos 6 x + cos 12 x + 3 cos 10 x + 3 cos 8 x is equal to

Answer 1

0

Answer 2

Given:

$$\sin x + \sin 2x = 1$$

Use the double-angle identity $\sin 2x = 2 \sin x \cos x$. Then,

$$\sin x (1+2\cos x)=1.$$

A simple value that works is $x=\frac{\pi}{2}$ because:

$$\sin \frac{\pi}{2}=1,\quad \cos\frac{\pi}{2}=0 \quad\Rightarrow\quad 1\cdot (1+0)=1.$$

Now, substitute $x=\frac{\pi}{2}$ in the expression:

$$\cos6x+\cos12x+3\cos10x+3\cos8x.$$

Compute each term:

• $\cos6x = \cos\left(6\cdot\frac{\pi}{2}\right)=\cos(3\pi)=-1,$
• $\cos12x = \cos\left(12\cdot\frac{\pi}{2}\right)=\cos(6\pi)=1,$
• $\cos10x = \cos\left(10\cdot\frac{\pi}{2}\right)=\cos(5\pi)=-1,$
• $\cos8x = \cos\left(8\cdot\frac{\pi}{2}\right)=\cos(4\pi)=1.$

Substitute these back:

$$-1 + 1 + 3(-1) + 3(1) = 0.$$