EXAMPLE 34 Prove that tan-1 V + *
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EXAMPLE 34 Prove that tan-1 V + *

11th - 12th Class
Maths
Solution
96
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SOUTION Putting ( x=tan theta, ) we get [ begin{aligned} tan ^{-1}left(frac{sqrt{1+x^{2}}-1}{x}right) &=tan ^{-1}left(frac{sqrt{1+tan ^{2} theta}-1}{tan theta}right) &=tan ^{-1}left(frac{sec theta-1}{tan theta}right)=tan ^{-1}left(frac{1-cos theta}{sin theta}right) &=tan ^{-1}left{frac{2 sin ^{2}left(frac{theta}{2}right)}{2 sin left(frac{theta}{2}right) cos left(frac{theta}{2}right)}right}=tan ^{-1}left(tan frac{theta}{2}right) therefore quad tan ^{-1}left(frac{sqrt{1+x^{2}}-1}{x}right) &=frac{theta}{2}=frac{1}{2} tan ^{-1} x end{aligned} ]
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