QPaperGen

Step 1: Expand the left-hand side
\begin{align*}
&(\cos x - \cos y)^2 + (\sin x - \sin y)^2 \\
&= \cos^2 x + \cos^2 y - 2\cos x \cos y + \sin^2 x + \sin^2 y - 2\sin x \sin y
\end{align*}

Step 2: Group terms
\[
= (\cos^2 x + \sin^2 x) + (\cos^2 y + \sin^2 y) - 2(\cos x \cos y + \sin x \sin y)
\]

Step 3: Use identities
\[
\cos^2 x + \sin^2 x = 1,\quad \cos^2 y + \sin^2 y = 1
\]
\[
\Rightarrow \text{LHS} = 1 + 1 - 2(\cos x \cos y + \sin x \sin y) = 2 - 2\cos(x - y)
\]

Step 4: Use the identity
\[
\cos(x - y) = 1 - 2\sin^2\left(\frac{x - y}{2}\right)
\]

Substitute:
\begin{align*}
\text{LHS} &= 2 - 2\left(1 - 2\sin^2\left(\frac{x - y}{2}\right)\right) \\
&= 2 - 2 + 4\sin^2\left(\frac{x - y}{2}\right) \\
&= 4\sin^2\left(\frac{x - y}{2}\right)
\end{align*}

Therefore,
\[
(\cos x - \cos y)^2 + (\sin x - \sin y)^2 = 4\sin^2\left(\frac{x - y}{2}\right) \qquad
\textbf{Hence Proved.}
\]