$ an \left\{\frac{1}{2} \cos ^{-1} \frac{\sqrt{5}}{3} ight\}$ has the value

Question

$	an \left\{\frac{1}{2} \cos ^{-1} \frac{\sqrt{5}}{3}ight\}$ has the value

Aicharya · Accepted Answer

Let $\frac{1}{2} \cos ^{-1} \frac{\sqrt{5}}{3}=	heta$ $\cos 2 	heta=\frac{\sqrt{5}}{3}$ Now $	an \left\{\frac{1}{2} \cos ^{-1} \frac{\sqrt{5}}{3}ight\}$ $\begin{aligned} & =	an 	heta=\sqrt{\frac{1-\cos 2 	heta}{1+\cos 2 	heta}}=\sqrt{\frac{1-\frac{\sqrt{5}}{3}}{1+\frac{\sqrt{5}}{3}}}=\sqrt{\frac{3-\sqrt{5}}{3+\sqrt{5}}} \ & =\sqrt{\frac{3-\sqrt{5}}{3+\sqrt{5}} 	imes \frac{3-\sqrt{5}}{3+\sqrt{5}}}=\sqrt{\frac{(3-\sqrt{5})^2}{3^2-5}}=\frac{3-\sqrt{5}}{2} \end{aligned}$

$\tan \left\{\frac{1}{2} \cos ^{-1} \frac{\sqrt{5}}{3}\right\}$ has the value

Solution