Question

If $\sin A+\cos A=\sqrt{2}\sin (90-A)$, then obtain the value of $\cot A$.

Answer 1

Given

$sinA+cosA=\sqrt{2}sin(90-A)$

$\Rightarrow \frac{1}{\sqrt{2}}sinA+\frac{1}{\sqrt{2}}cosA=sin(90-A)$

$\Rightarrow sinA.cos{45}^{\circ }+cosA+sin{45}^{\circ }=sin(90-A)$

$\Rightarrow (A+{45}^{\circ })=sin({90}^{\circ }-A)$

Equating

$A+45=90-A$

$\Rightarrow 2A={45}^{\circ }$

$\Rightarrow A=22{\frac{1}{2}}^{\circ }$

$A=\frac{\pi }{8}$

$\therefore cotA=cot\frac{\pi }{8}=\sqrt{2}+1$