Question

$\text{If}cot\theta =\frac{1}{\sqrt{3}},\text{find the value of}\frac{1-co{s}^2\theta }{2-si{n}^2\theta }$

Answer 1

Given $cot\theta =\frac{1}{\sqrt{3}}$

We know that, $1+co{t}^2\theta =cose{c}^2\theta$

$\Rightarrow 1+\frac{1}{3}=cose{c}^2\theta$

$\Rightarrow cosec\theta =\frac{2}{\sqrt{3}}$

$\Rightarrow sin\theta =\frac{\sqrt{3}}{2}$

and $cos\theta =\sqrt{1-si{n}^2\theta }=\frac{1}{2}$

Now, $\frac{1-co{s}^2\theta }{2-si{n}^2\theta }=\frac{1-\frac{1}{4}}{2-\frac{3}{4}}=\frac{3}{5}$