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Mathematics
cos -1( (3+5 cos x/5+3 cos x) ) is equal to:
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Q. $ {{\cos }^{-1}}\left( \frac{3+5\cos x}{5+3\cos x} \right) $ is equal to:
KEAM
KEAM 2005
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) $
Solution:
$ {{\cos }^{-1}}\left( \frac{3+5\cos x}{5+3\cos x} \right) $
$ ={{\cos }^{-1}}\left\{ \frac{3+5\left( \frac{1-{{\tan }^{2}}x/2}{1+{{\tan }^{2}}x/2} \right)}{5+3\left( \frac{1-{{\tan }^{2}}x/2}{1+{{\tan }^{2}}x/2} \right)} \right\} $
$ ={{\cos }^{-1}}\left( \frac{4-{{\tan }^{2}}x/2}{4+{{\tan }^{2}}x/2} \right) $
$ ={{\cos }^{-1}}\left\{ \frac{1-{{\left( \frac{1}{2}\tan x/2 \right)}^{2}}}{1+{{\left( \frac{1}{2}\tan x/2 \right)}^{2}}} \right\} $
$ =2{{\tan }^{-1}}\left( \frac{1}{2}\tan \frac{x}{2} \right) $