If cos A + sin B = m and sin A + cos B = n, Prove that 2sin(A+B) = m² + n² − 2

Question

If

\[ \cos A+\sin B=m \]

and

\[ \sin A+\cos B=n \]

prove that:

\[ 2\sin(A+B)=m^2+n^2-2 \]

Proof

Given,

\[ \cos A+\sin B=m \]

\[ \sin A+\cos B=n \]

Squaring and adding,

\[ (\cos A+\sin B)^2+(\sin A+\cos B)^2 = m^2+n^2 \]

\[ \cos^2A+\sin^2B+2\cos A\sin B \]

\[ +\sin^2A+\cos^2B+2\sin A\cos B = m^2+n^2 \]

\[ (\sin^2A+\cos^2A) + (\sin^2B+\cos^2B) \]

\[ +2(\sin A\cos B+\cos A\sin B) = m^2+n^2 \]

\[ 1+1+2\sin(A+B) = m^2+n^2 \]

\[ 2+2\sin(A+B) = m^2+n^2 \]

\[ 2\sin(A+B) = m^2+n^2-2 \]

Hence proved.