Question

$\mathrm{If}\cot \theta =\frac{1}{\sqrt{3}},\mathrm{then}\mathrm{write}\mathrm{the}\mathrm{value}\mathrm{of}\frac{\left(1-{\cos }^2\theta \right)}{\left(2-{\sin }^2\theta \right)}.$

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}, \\ \cot \theta =\frac{1}{\sqrt{3}} \\ \Rightarrow \cot \theta =\cot \left(\frac{\mathrm{\pi }}{3}\right) \\ \Rightarrow \theta =\frac{\mathrm{\pi }}{3} \\ \mathrm{Now}, \\ \frac{\left(1-{\cos }^2\theta \right)}{\left(2-{\sin }^2\theta \right)} \\ =\frac{1-{\cos }^2\left({\displaystyle \frac{\mathrm{\pi }}{3}}\right)}{2-{\sin }^2\left({\displaystyle \frac{\mathrm{\pi }}{3}}\right)} \\ =\frac{1-{\left({\displaystyle \frac{1}{2}}\right)}^2}{2-{\left({\displaystyle \frac{\sqrt{3}}{2}}\right)}^2} \\ =\frac{\left({\displaystyle \frac{1}{1}}-{\displaystyle \frac{1}{4}}\right)}{\left({\displaystyle \frac{2}{1}}-{\displaystyle \frac{3}{4}}\right)} \\ =\frac{\left({\displaystyle \frac{3}{4}}\right)}{\left({\displaystyle \frac{5}{4}}\right)} \\ =\frac{3}{5} \end{gathered}$$