Question

$\frac{\cos {135}^{\circ }-\cos {120}^{\circ }}{\cos {135}^{\circ }+\cos {120}^{\circ }}=(3-2\sqrt{2})$

Answer 1

$\frac{\cos {135}^{\circ }-\cos {120}^{\circ }}{\cos {135}^{\circ }+\cos {120}^{\circ }}=\frac{\cos (180-45)-\cos (90+30)}{\cos (180-45)+\cos (90+30)}$

$=\frac{-\cos {45}^{\circ }+\sin {30}^{\circ }}{-\cos {45}^{\circ }-\sin {30}^{\circ }}$

$=\frac{-\frac{1}{\sqrt{2}}+\frac{1}{2}}{-\frac{1}{\sqrt{2}}-\frac{1}{2}}=\frac{-2+\sqrt{2}}{-2-\sqrt{2}}\times \frac{-2+\sqrt{2}}{-2+\sqrt{2}}$

$=\frac{(-2+\sqrt{2}{)}^2}{4-2}=3-2\sqrt{2}$