Prove that : `2 tan^(-1) (cosec tan^(-1) x - tan cot^(-1) x) = tan^(-1) x`

Correct option is: A. cos θ sin θ

Correct Answer - A `cosec (cosec^(-1) x) = x AA x in R -(-1, 1)` Also range of `cosec^(-1) (cosec x) in [-(pi)/(2), 0) uu (0, (pi)/(2]` So combing these two,...

Correct Answer - A Let `sqrt(tan alpha) = tan x`. Then `u = cot^(-1) (tan x) - tan^(-1) (tan x)` `= (pi)/(2) - x -x = (pi)/(2) - 2x` or `2x...

Correct Answer - A::B::C::D Since `|tan^(-1)x| = {(tan^(-1) x," if " x ge 0),(-tan^(-1) x," if " x lt 0):}` `rArr |tan^(-1)x| = tan^(-1) |x| AA x in R` `rArr tan...

Correct Answer - A::B::D Let `t_(r)` denote the `rth` term of the series 3, 7, 13, 21, ... and `{:(S = 3 + 7 + 13 + 21 + ...+ t_(n)),(S...

Correct Answer - 5 `(cot^(-1) x) (tan^(-1)x) + (2 -(pi)/(2)) cot^(-1) x - 3 tan^(-1) x 3 (2 - (pi)/(2)) gt 0` `rArr cot^(-1) x (tan^(-1) x -(pi)/(2)) + 2 cot^(-1)...

Correct Answer - A `cot.(A)/(2) + cot.(B)/(2) + cot.(C)/(2)` `= (s(s-a))/(Delta) + (s(s-b))/(Delta) + (s-(s -c))/(Delta)` `= (s)/(Delta) [3s - (a + b + c)] = (s^(2))/(Delta) = ((Delta//r)^(2))/(Delta) = (Delta)/(r^(2))`

Correct Answer - B::D Given `(s(s-b))/(Delta) + ((s-c))/(Delta) = (2s(s-a))/(Delta)` `rArr s-b + s - c = 2 (s-a)` `rArr b+c = 2a` So, locus of vertex A is an ellipse

Correct answer is (C) \(\frac{x^2 + 1}{2x}\)

Correct answer is (D) tan A