Q. If $\sin{A + cos\ B = a}$ and $\sin{B + \cos{A = b}}$, then $\sin{(A + B)}$ is equal to
A) $\frac{a^{2} + b^{2}}{2}$
B) $\frac{a^{2} - b^{2} + 2}{2}$
C) $\frac{a^{2} + b^{2} - 2}{2}$
D) None of these
Explanation
(c)
On squaring and adding the given equations, we get
$\sin^{2}{A + cos^{2}B + 2\sin{Acos\ B + \sin^{2}{B + \cos^{2}{A + 2\sin{B\cos{A = a^{2} + b^{2}}}}}}}$
$\Rightarrow \ 2\sin{(A + B) + 2 = a^{2} + b^{2}}$
$\Rightarrow \ \sin{(A + B) = \frac{a^{2} + b^{2} - 2}{2}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.