Question

Evaluate the Following

${\displaystyle \frac{4}{{\cot }^2{30}^{\circ }}+\frac{1}{{\sin }^2{60}^{\circ }}-{\cos }^2{45}^{\circ }}$

Answer 1

${\displaystyle \frac{4}{{\cot }^2{30}^{\circ }}+\frac{1}{{\sin }^2{60}^{\circ }}-{\cos }^2{45}^{\circ }}$ .....(i)

By trigonometric ratios we have

${\displaystyle {\cot 30}^{\circ }=\sqrt{3} {\sin 60}^{\circ }=\frac{\sqrt{3}}{2} {\cos 45}^{\circ }=\frac{1}{\sqrt{2}}}$

By substituting above values in (i), we get

${\displaystyle \frac{4}{{\left(\sqrt{3}\right)}^2}+\frac{1}{{\left(\frac{\sqrt{3}}{2}\right)}^2}-{\left(\frac{1}{\sqrt{2}}\right)}^2}$

${\displaystyle \frac{4}{3}+\frac{4}{3}-\frac{1}{2}=\frac{13}{6}}$