Question

Solve: $\sin {105}^o+\cos {105}^o$

Answer 1

We have,

$\sin {105}^{\circ }+\cos {105}^{\circ }$

$\Rightarrow \sqrt{2}(\frac{1}{\sqrt{2}}\sin {105}^{\circ }+\frac{1}{\sqrt{2}}\cos {105}^{\circ })$

$\Rightarrow \sqrt{2}(\sin {45}^{\circ }\sin {105}^{\circ }+\cos {45}^{\circ }\cos {105}^{\circ })$

$\Rightarrow \sqrt{2}(\cos {105}^{\circ }\cos {45}^{\circ }+\sin {105}^{\circ }\sin {45}^{\circ })$

$\Rightarrow \sqrt{2}\cos ({105}^{\circ }-{45}^{\circ })[\because \cos (A-B)=\cos A\cos B+\sin A\sin B]$

$\Rightarrow \sqrt{2}\cos {60}^{\circ }$

$\Rightarrow \sqrt{2}\times \frac{1}{2}$

$\Rightarrow \frac{1}{\sqrt{2}}$

Hence, this is the answer.