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sourtoN Let ( tan ^{-1} frac{1}{x}= ) Q. Then, ( frac{1}{x}=tan theta Rightarrow x=cot theta )
[
begin{array}{l}
begin{aligned}
therefore quad mathrm{LHS} &=2 tan ^{-1} frac{1}{x}=2 mathrm{Q}
mathrm{RHS} &=sin ^{-1}left(frac{2 cot theta}{cot ^{2} theta+1}right)=sin ^{-1}left(frac{2 tan theta}{1+tan ^{2} theta}right)
&=sin ^{-1}(sin 2 theta)=2 mathrm{Q}
end{aligned}
therefore quad text { LHS }=mathrm{RHS}
text { Hence, } 2 tan ^{-1} frac{1}{x}=sin ^{-1}left(frac{2 x}{x^{2}+1}right)
end{array}
] 
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EXAMPLE 39 Prove that 2 tan-1- = sin -1
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