1. Tardigrade
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  3. Mathematics
  4. cot -1(√ cos α )= tan -1(√ cos α )=x, then sin x is equal to:

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Q. $ {{\cot }^{-1}}(\sqrt{\cos \alpha })={{\tan }^{-1}}(\sqrt{\cos \alpha })=x, $ then $ \sin x $ is equal to:

JamiaJamia 2005

Solution:

$ {{\cot }^{-1}}(\sqrt{\cos \alpha })-{{\tan }^{-1}}(\sqrt{\cos \alpha })=x $ $ {{\tan }^{-1}}\left( \frac{1}{\sqrt{\cos }\alpha } \right)-{{\tan }^{-1}}(\sqrt{\cos \alpha })=x $ $ \Rightarrow $ $ {{\tan }^{-1}}\frac{\frac{1}{\sqrt{\cos }\alpha }-\sqrt{\cos \alpha }}{1+\frac{1}{\sqrt{\cos \alpha }}.\sqrt{\cos \alpha }}=x $ $ \Rightarrow $ $ {{\tan }^{-1}}\frac{1-\cos \alpha }{2\sqrt{\cos }\alpha }=x $ $ \Rightarrow $ $ \tan x=\frac{1-\cos \alpha }{2\sqrt{\cos \alpha }} $ $ \sin x=\frac{1-\cos \alpha }{1+\cos \alpha } $ $ =\frac{1-(1-2{{\sin }^{2}}\alpha /2)}{1+2{{\cos }^{2}}\alpha /2-1} $ $ \sin x={{\tan }^{2}}\frac{\alpha }{2} $