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