InterviewSolution
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InterviewSolution
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Prove that `2 tan^(-1) (cosec tan^(-1) x - tan cot^(-1) x) = tan^(-1) x (x != 0)` |
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Answer» Case I: `x gt 0` 2 tan^(-1) (cosec tan^(-1) x - tan cot^(-1) x)` `= 2 tan^(-1) `(cosec(cosec^(-1) (sqrt(1 + x^(2)))/(x)) - tan (tan^(-1). (1)/(x)))` `= 2 tan ^(-1) ((sqrt(1 + x^(2)))/(x) - (1)/(x))` `= 2 tan^(-1) ((sec theta - 1)/(sin theta))` (Putting `x = tan theta`) `= 2 tan^(-1) ((1 - cos theta)/(sin theta))` `= 2 tan^(-1) ((2 sin^(2) theta//2)/(2 sin theta//2 cos theta//2))` `= 2 tan^(-1) (tan.(theta)/(2)) = 2 xx (theta)/(2)` `= theta = tan^(-1) x` Case II: `x lt 0` Let `y = -x` `:. y gt 0` `:. 2 tan^(-1) (cosec tan^(-1) x - tan cot^(-1) x)` `= 2 tan^(-1) (cosec tan^(-1) (-y) - tan cot^(-1) (-y))` `=2 tan^(-1) (cosec (-tan^(-1) y) - tan (pi - cot^(-1) y))` `= 2 tan^(-1) (-cosec (tan^(-1) y) + tan (cot^(-1) y))` `= 2 tan^(-1) (-cosec (cosec^(-1) (sqrt(1 + y^(2)))/(y)) + tan (tan^(-1).(1)/(y)))` `= 2 tan^(-1) (-(sqrt1 + y^(2))/(y) + (1)/(y))` `= 2 tan^(-1) ((sqrt(1 + y^(2))-1)/(y))` `= -tan^(-1) y` `= tan^(-1) (-y)` `= tan^(-1) x` |
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