If tan θ=1√2find the value of cosec2θ−sec2θcosec2θ+cot2

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

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Given, $t a n θ = \frac{1}{\sqrt{2}}$

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

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

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

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

So, $s i n θ = \sqrt{1 - c o s^{2} θ} = \sqrt{1 - \frac{2}{3}} = \frac{1}{\sqrt{3}}$

$\Rightarrow c o s e c θ = \sqrt{3}$

Also, $c o t θ = \frac{1}{t a n θ} = \sqrt{2}$

Now, $\frac{c o s e c^{2} θ - s e c^{2} θ}{c o s e c^{2} θ + c o t^{2} θ} = \frac{3 - \frac{3}{2}}{3 + 2} = \frac{3}{10}$