Question

Evaluate each of the following :
$\frac{1}{co{t}^2{30}^{\circ }}+\frac{1}{si{n}^2{60}^{\circ }}-co{s}^2{45}^{\circ }$

Answer 1

Given $\frac{1}{co{t}^2{30}^{\circ }}+\frac{1}{si{n}^2{60}^{\circ }}-co{s}^2{45}^{\circ }$

$=ta{n}^2{30}^{\circ }+cose{c}^2{60}^{\circ }-co{s}^2{45}^{\circ }$

$={\left(\frac{1}{\sqrt{3}}\right)}^2+{\left(\frac{2}{\sqrt{3}}\right)}^2+{\left(\frac{1}{\sqrt{2}}\right)}^2$ $[\because cosec{60}^{\circ }=\frac{2}{\sqrt{3}},\cos {45}^{\circ }=\frac{1}{\sqrt{2}},\tan {30}^{\circ }=\frac{1}{\sqrt{3}}]$

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

$=\left(\frac{1+4}{3}\right)-\frac{1}{2}$

$=\frac{5}{3}-\frac{1}{2}$

$=\frac{10-3}{6}$

$=\frac{7}{6}$