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Question

Solve the following equation for x:

tan11x1+x=12tan1x,(x>0).

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Solution

Given,
tan1(1x1+x)=12tan1x or 2tan1(1x1+x)=tan1xtan12(1x1+x)1(1x1+x)2=tan1x [ 2tan1y=tan1(2y1y2)]tan12(1x)(1+x)(1+x)2(1x)2(1+x)2=tan1x[2(1x)(2+x)(1+x)2(1x)2]=tan1xtan1[2(1x)212+x2+2x12x2+2x]=tan1xtan1[2(1x2)4x]=tan1xtan1(2x22x)=tan1x1x22x=x1x2=2x21=3x2x2=13x=±13
[ x>0 given, so we do not take x=13]
x=13


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