Question

Solve:

$\frac{\sin {75}^o+\sin {15}^o}{\cos {75}^o+\cos {15}^o}=$

Answer 1

$\frac{\sin {75}^o+\sin {15}^o}{\cos {75}^o+\cos {15}^o}=?$

$\Rightarrow \frac{sin({45}^{\circ }+{30}^{\circ })+sin({45}^{\circ }-{30}^{\circ })}{cos({45}^{\circ }+{30}^{\circ })+cos({45}^{\circ }-{30}^{\circ })}$

$\therefore sin(a\pm b)=sinacosb\pm cosasinb]$

$\Rightarrow \frac{(sin{45}^{\circ }cos{30}^{\circ }+cos{45}^{\circ }sin{30}^{\circ })+(sin{45}^{\circ }cos{30}^{\circ }-cos{45}^{\circ }sin{30}^{\circ })}{(cos{45}^{\circ }cos{30}^{\circ }-sin{45}^{\circ }sin{30}^{\circ })}$

$=\frac{(\frac{1}{\sqrt{2}}.\frac{\sqrt{3}}{2}+\frac{1}{\sqrt{2}}.\frac{1}{2})+(\frac{1}{\sqrt{2}}.\frac{\sqrt{3}}{2}-\frac{1}{\sqrt{2}}.\frac{1}{2})}{(\frac{1}{\sqrt{2}}.\frac{\sqrt{3}}{2}+\frac{1}{\sqrt{2}}.\frac{1}{2})+(\frac{1}{\sqrt{2}}.\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}.\frac{1}{2})}$

$=\frac{(\frac{\sqrt{3}}{2\sqrt{2}}+\frac{1}{2\sqrt{2}})+(\frac{\sqrt{3}}{2\sqrt{2}}-\frac{1}{2\sqrt{2}})}{\left(\frac{\sqrt{3}}{2\sqrt{2}}\right)+(\frac{\sqrt{3}}{2}-\frac{1}{2\sqrt{2}})}$

$=\frac{(\sqrt{3}+1)+(\sqrt{3}-1)}{(\sqrt{3}+1)+(\sqrt{3}-1)}$

$\Rightarrow \frac{\sqrt{3}+1+\sqrt{3}-1}{\sqrt{3}+1\sqrt{3}-1}\Rightarrow 2\sqrt{3}\Rightarrow 1$