A) \[{{\tan }^{-1}}\left( \frac{1}{2}\tan \frac{x}{2} \right)\]
B) \[2{{\tan }^{-1}}\left( 2\tan \frac{x}{2} \right)\]
C) \[\frac{1}{2}{{\tan }^{-1}}\left( 2\tan \frac{x}{2} \right)\]
D) \[2{{\tan }^{-1}}\left( \frac{1}{2}\tan \frac{x}{2} \right)\]
E) \[{{\tan }^{-1}}\left( \tan \frac{x}{2} \right)\]
Correct Answer: D
Solution :
We take \[x=\frac{\pi }{2}\] then \[\cos x=0\] \[{{\cos }^{-1}}\left( \frac{3+5\cos x}{5+3\cos x} \right)={{\cos }^{-1}}\left( \frac{3}{5} \right)\] \[={{\tan }^{-1}}\left( \frac{4}{3} \right)\] Put \[x=\frac{\pi }{2}\] in \[2{{\tan }^{-1}}\left( \frac{1}{2}\tan \frac{x}{2} \right)\] we get \[2{{\tan }^{-1}}\left( \frac{1}{2}\tan \frac{\pi }{4} \right)\] \[=2{{\tan }^{-1}}\left( \frac{1}{2} \right)={{\tan }^{-1}}\left( \frac{2.\frac{1}{2}}{1-\frac{1}{4}} \right)\]\[={{\tan }^{-1}}\left( \frac{4}{3} \right)\].
You need to login to perform this action.
You will be redirected in
3 sec
