Prove $\sin((n+1)x) \sin((n+2)x)+\cos((n+1)x) \cos((n+2)x)=\cos(x)$

Question

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

Hi, I have been trying to solve this trigonometric function since last hour but not able to please help me to solve the above trigonometric function.

Answer 1

We use the subtraction rule for cosines: $$\cos(a-b)=\cos a\cos b+\sin a\sin b.\tag{$1$}$$ Put $a=(n+2)x$ and $b=(n+1)x$. The right-hand side of $(1)$ is then the complicated expression you were given, except yours mentioned the sines first.

So your expression is equal to $\cos\left((n+2)x-(n+1)x\right)$. But $(n+2)x-(n+1)x=x$.