Prove that:

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

Q: Prove that: \[ \sin(n+1)x \cdot \sin(n+2)x + \cos(n+1)x \cdot \cos(n+2)x = \cos x \]

Explanation: \begin{align*} \text{LHS} &= \sin(n+1)x \cdot \sin(n+2)x + \cos(n+1)x \cdot \cos(n+2)x \\ &= \cos\left[ (n+1)x - (n+2)x \right] \\ &= \cos(-x) = \cos x = \text{RHS} \qquad \textbf{Hence proved.} \end{align*}

Question

Prove that:

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

Accepted Answer

Explanation: 
\begin{align*}
	ext{LHS} &= \sin(n+1)x \cdot \sin(n+2)x + \cos(n+1)x \cdot \cos(n+2)x \
&= \cos\left[ (n+1)x - (n+2)x ight] \
&= \cos(-x) = \cos x = 	ext{RHS} \qquad 	extbf{Hence proved.}
\end{align*}

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