Question

The value of $\tan \left(\frac{1}{2} \cot ^{-1}(3)\right)$ equals

• A. $(3+\sqrt{10})^{-1}$
• B. $(10+\sqrt{3})^{-1}$
• C. $(3+\sqrt{10})$
• D. $(10+\sqrt{3})$

Answer 1

Answer: (A)

Let $\cot ^{-1} 3=\theta \Rightarrow \cot \theta=3 \Rightarrow \cos \theta=\frac{3}{\sqrt{10}}$
now $\tan ^2\left(\frac{\theta}{2}\right) =\frac{1-\cos \theta}{1+\cos \theta}=\frac{1-\frac{3}{\sqrt{10}}}{1+\frac{3}{\sqrt{10}}}=\frac{\sqrt{10}-3}{\sqrt{10}+3}=\frac{1}{(\sqrt{10}+3)^2} $
$\tan \left(\frac{\theta}{2}\right) \left.=(10+\sqrt{3})^{-1} \text { as } \tan \left(\frac{\theta}{2}\right)>0\right]$