Question

Prove:
${(\cos A+\cos B)}^2+{(\sin A+\sin B)}^2=4{\cos }^2\left(\frac{A-B}{2}\right)$

Answer 1

LHS=${\cos }^{2}A+{\cos }^{2}B+2\cos A\cos B+{\sin }^A+{\sin }^{@}B+2\sin A\sin B$

$=2+2(\cos A\cos B+\sin A\sin B)$

$=2(1+\cos (A-B\left)\right)$

$=2\ast 2{\cos }^2\frac{A-B}{2}$

$=RHS$