Mark the tick against the correct answer in the following:

Question:

Mark the tick against the correct answer in the following:

$\tan \frac{1}{2}\left(\cos ^{-1} \frac{\sqrt{5}}{3}\right)=?$

A. $\frac{(3-\sqrt{5})}{2}$

B. $\frac{(3+\sqrt{5})}{2}$

C. $\frac{(5-\sqrt{3})}{2}$

D. $\frac{(5+\sqrt{3})}{2}$

 

Solution:

To Find: The value of $\tan \frac{1}{2}\left(\cos ^{-1} \frac{\sqrt{5}}{3}\right)$

Let, $x=\cos ^{-1} \frac{\sqrt{5}}{3}$

$\Rightarrow \cos x=\frac{\sqrt{5}}{3}$

Now, $\tan \frac{1}{2}\left(\cos ^{-1} \frac{\sqrt{5}}{3}\right)$ becomes

$\tan \frac{1}{2}\left(\cos ^{-1} \frac{\sqrt{5}}{3}\right)=\tan \frac{1}{2}(x)=\tan \frac{x}{2}$

$=\sqrt{\frac{1-\cos x}{1+\cos x}}$

$=\sqrt{\frac{1-\left(\frac{\sqrt{5}}{3}\right)}{1+\frac{\sqrt{5}}{2}}}$

$=\sqrt{\frac{3-\sqrt{5}}{3+\sqrt{5}}}$

$=\sqrt{\frac{3-\sqrt{5}}{3+\sqrt{5}}} \times \sqrt{\frac{3-\sqrt{5}}{3-\sqrt{5}}}$

$\tan \frac{1}{2}\left(\cos ^{-1} \frac{\sqrt{5}}{3}\right)=\frac{3-\sqrt{5}}{2}$

 

Leave a comment

Close

Click here to get exam-ready with eSaral

For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.

Download Now