Prove that tan " Var = LC05-1 (1-x)...
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Prove that tan " Var = LC05-1 (1-x),x € [0, 1] 1 + x

11th - 12th Class
Maths
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( x=tan ^{2} theta, ) we get soution Putting [ begin{aligned} R H S &=frac{1}{2} cos ^{-1}left(frac{1-x}{1+x}right) &=frac{1}{2} cos ^{-1}left(frac{1-tan ^{2} theta}{1+tan ^{2} theta}right) &=frac{1}{2} cos ^{-1}(cos 2 theta) =&left(frac{1}{2} times 2 thetaright)=theta=tan ^{-1} sqrt{x}=mathrm{LHS} &left[because x=tan ^{2} theta Rightarrow tan theta=sqrt{x} Rightarrow theta=tan ^{-1} sqrt{x}right] text { Hence, } tan ^{-1} sqrt{x}=frac{1}{2} cos ^{-1}left(frac{1-x}{1+x}right) & end{aligned} ]
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