Question

Find : $\sin ({45}^o+A).\sin ({45}^o-A)-\frac{1}{2}\cos 2A$

Answer 1

$\sin ({45}^o+A)\sin ({45}^o-A)-\frac{1}{2}\cos 2A$

$\because {\sin }^{2}A-{\sin }^{2}B=\sin (A+B)\sin (A-B)$

$\Rightarrow {\sin }^2{45}^o-{\sin }^{2}A-\frac{1}{2}(1-2{\sin }^{2}A)$

$\because \cos 2\theta =1-2{\sin }^2\theta$

$\sin {45}^o=\frac{1}{\sqrt{2}}$

$\Rightarrow \frac{1}{2}-{\sin }^{2}A-\frac{1}{2}+{\sin }^{2}A$

$=0$.