InterviewSolution
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InterviewSolution
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Let `tan^(-1)y=tan^(-1)x+tan^(-1)((2x)/(1-x^2))`, where `|x| |
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Answer» Correct Answer - A Given ` tan ^(-1) y = tan ^(-1) x + tan ^(-1) ((2x)/( 1- x ^(2)))` where `|x| lt (1)/(sqrt 3) rArr tan ^(-1) y = tan ^(-1) {( x + (2x )/( 1-x ^(2)))/( 1- x (( 2x )/( 1 - x ^(2)))) }` `[because tan ^(-1) x + tan ^(-1) y = tan ^(-1) ((x + y )/( 1- xy )), where x gt 0, y gt 0 and xy lt 1]` `" " =tan ^(-1) ((x - x ^(3) + 2x )/( 1 - x ^(2) - 2x ^(2))) ` `tan ^(-1) y = tan ^(-1) ((3x - x ^(3))/( 1 - 3x ^(2)))` `rArr " " y = ( 3x - x ^(3))/( 1 - 3x ^(2))` ` " " |x| lt (1)/(sqrt 3)` `rArr " " - (1)/(sqrt (3)) lt x lt (1)/(sqrt 3)` Let ` x = tan theta ` `rArr " " - ( pi )/(6) lt theta lt (pi)/(6)` `therefore " " tan ^(-1) y = theta + tan ^(-1) ( tan 2 theta) = theta + 2theta = 3 theta ` `rArr " " y = tan 3 theta ` `rArr " " y = ( 3 tan theta - tan ^(3) theta)/( 1- 3 tan ^(2) theta)` ` rArr " " y = ( 3x - x ^(3))/( 1- 3x ^(2))` |
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