Question

Show that $\frac{\cos {15}^o-\sin {15}^o}{\cos {15}^o+\sin {15}^o}=\frac{1}{\sqrt{3}}$

Answer 1

$\begin{array}{c}\frac{\cos {15}^o-\sin {15}^o}{\cos {15}^o+\sin {15}^o}\\ =\frac{{\left(\cos {15}^o-\sin {15}^o\right)}^2}{\left(\cos {15}^o+\sin {15}^o\right)\left(\cos {15}^o-\sin {15}^o\right)}\\ =\frac{{\cos }^2{15}^o+{\sin }^2{15}^o-2\cos {15}^o\sin {15}^o}{\left({\cos }^2{15}^o-{\sin }^2{15}^o\right)}\\ =\frac{1-\sin \left(2\left(15\right)\right)}{\cos \left(2\left(15\right)\right)}\\ =\frac{1-\sin \left(30\right)}{\cos \left(30\right)}\\ =\frac{1-\frac{1}{2}}{\frac{\sqrt{3}}{2}}=\frac{1/2}{\sqrt{3}/2}=\frac{1}{\sqrt{3}}\end{array}$