Question

If the value of $\cot \left({11\frac{1}{4}}^o\right)+\tan \left({112\frac{1}{2}}^o\right)-\cot \left({112\frac{1}{2}}^o\right)-\tan \left({11\frac{1}{4}}^o\right)$ equals $\sqrt{n}$, where $n\in N$ then find the value of $n$

Answer 1

We know that,

$cotA-tanA=2cot2A$,

Using this identity , we get

$⟹cot\left(\frac{{45}^o}{4}\right)-tan\left(\frac{{45}^o}{4}\right)-cot\left(\frac{{225}^o}{2}\right)+tan\left(\frac{{225}^o}{2}\right)$

$⟹2cot(2\times \frac{{45}^o}{4})-2cot(2\times \frac{{225}^o}{2})$

$⟹2\left[cot\right({22.5}^o)-cot({225}^o\left)\right]$

$⟹2[\sqrt{2}+1-1]$

$⟹2\sqrt{2}=\sqrt{8}$

Hence, $n=8$.