QPaperGen

\(\textbf{Step 1: }\)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*}

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

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

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

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

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

\(\textbf{Hence Proved.}\)