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  4. The value of sin ( sin -1(1/3)+ sec -13 )+ cos ( tan -1(1/2)+ tan -12 ) is

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Q. The value of $ \sin \left( {{\sin }^{-1}}\frac{1}{3}+{{\sec }^{-1}}3 \right)+ $ $ \cos \left( {{\tan }^{-1}}\frac{1}{2}+{{\tan }^{-1}}2 \right) $ is

JamiaJamia 2006

Solution:

Key Idea: $ {{\sin }^{-1}}x+{{\cos }^{-1}}x=\frac{\pi }{2} $ and $ {{\tan }^{-1}}x+{{\cot }^{-1}}x=\frac{\pi }{2} $ $ \sin \left( {{\sin }^{-1}}\frac{1}{3}+{{\sec }^{-1}}3 \right) $ $ +\cos \left( {{\tan }^{-1}}\frac{1}{2}+{{\tan }^{-1}}2 \right) $ $ =\sin \left( {{\sin }^{-1}}\frac{1}{3}+{{\cos }^{-1}}\frac{1}{3} \right) $ $ +\cos \left( {{\tan }^{-1}}\frac{1}{2}+{{\cot }^{-1}}\frac{1}{2} \right) $ $ =\sin \left( \frac{\pi }{2} \right)+\cos \left( \frac{\pi }{2} \right)=1 $