If cot θ=1√3find the value of 1−cos2θ2−sin2θ Given cot

$If c o t θ = \frac{1}{\sqrt{3}}, find the value of \frac{1 - c o s^{2} θ}{2 - s i n^{2} θ}$

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Given $c o t θ = \frac{1}{\sqrt{3}}$

We know that, $1 + c o t^{2} θ = c o s e c^{2} θ$

$\Rightarrow 1 + \frac{1}{3} = c o s e c^{2} θ$

$\Rightarrow c o s e c θ = \frac{2}{\sqrt{3}}$

$\Rightarrow s i n θ = \frac{\sqrt{3}}{2}$

and $c o s θ = \sqrt{1 - s i n^{2} θ} = \frac{1}{2}$

Now, $\frac{1 - c o s^{2} θ}{2 - s i n^{2} θ} = \frac{1 - \frac{1}{4}}{2 - \frac{3}{4}} = \frac{3}{5}$