If sin (A - B) = \(\frac{1}{2}\) and cos (A + B) = \(\frac{1}{2}\) , what is the value of sin A. sin 2B + sin² A . sin B cos B?

 \(\frac{4 + \sqrt {2 }}{8\sqrt {2 }}\)

sin (A - B) = \(\frac{1}{2}\) ⇒ sin (A - B) = sin 30°

A - B = 30°  ...........(i)

cos (A + B) =  \(\frac{1}{2}\) ⇒ cos (A + B) = cos 60°

A + B = 60° ............(ii)

Adding (i) , (ii)

2A = 90° ⇒ A = 45°

and B = 15°

sin A. sin 2B + sin² A. sin B cos B

= sin 45° sin 30° + (sin² 45°) × \(\frac{1}{2}\) × (2sin 15°.cos 15°)

= \(\frac{1}{\sqrt {2 }}\) × \(\frac{1}{2}\) + \(\frac{1}{2}\) × \(\frac{1}{(\sqrt {2 }})^2\) (sin 30°)

= \(\frac{1}{2\sqrt { 2}}\) + \(\frac{1}{4}\) × \(\frac{1}{2}\) = \(\frac{4 + \sqrt {2 }}{8\sqrt {2 }}\)