Evaluate: $$\cos\theta\cos2\theta\cos3\theta + \cos2\theta\cos3\theta\cos4\theta + …$$ upto $n$ terms
I tried solving the general term $\cos n\theta\cos (n+1)\theta\cos (n+2)\theta$.First, I applied the formula $2\cos\alpha\cos\beta = \cos(\alpha+\beta)+\cos(\alpha-\beta)$ on the two extreme terms. After solving I applied this once again and after further solving arrived at $$\frac{1}{4}[\cos(3n+3)\theta + \cos(n+1)\theta+\cos(n+3)\theta+\cos(n-1)\theta]$$
which I simplified to
$$\frac{\cos n\theta}{2}[\cos\theta+\cos(2n+3)\theta]$$
After this I am stuck as to what else I could do so as to make the telescope or something else to easily calculate the sum using some fact from trigonometry. Or maybe this is a dead end. And help or hints would be appreciated, thanks
Best Answer
$$\cos(n-1)t\cdot\cos nt\cdot\cos(n+1)t$$
$$=\dfrac{\cos nt(\cos2t+\cos2n t)}2$$
$$=\dfrac{\cos2t\cos nt}2+\dfrac{\cos nt+\cos3nt}4$$
$$=\dfrac{2\cos2t+1}4\cdot\cos nt+\dfrac{\cos3nt }4$$
Use $\sum \cos$ when angles are in arithmetic progression