Find the principal value of the following (i) `sin^(-1).(1)/(2)` (ii) `tan ^(-1).(1)/sqrt(3)` (iii) `cot ^(-1)(-sqrt3)`

Correct option is: C) cos \(\theta\)  + sin  \(\theta\) \(\frac{cos\theta}{1-tan\theta} + \frac{sin\theta}{1-cot\theta}\)= \(\frac {cos\theta}{\frac {1-sin\theta}{cos\theta}} + \frac {sin\theta}{\frac {1-cos\theta}{sin\theta}}\) = \(\frac {cos^2\theta}{cos\theta - sin \theta} + \frac {sin^2\theta}{sin\theta - cos\theta}\) = \(\frac {cos^2\theta}{cos\theta - sin \theta} - \frac {sin^2\theta}{sin\theta...

(i) Since `1 in [-pi//2, pi//2], sin^(-1) (sin 1) = 1` (ii) Since `2 in [-pi//2, pi//2], sin^(-1) (sin 2) != 2` `:. Sin^(-1) (sin 2) = sin^(-1) (sin (pi)...

`E = sqrt(sin^(-1)x_(1)) sqrt(cos^(-1) x_(2)) + sqrt(sin^(-1) x_(2)) sqrt(cos^(-1) x_(3)) + sqrt(sin^(-1) x_(3)) sqrt(cos^(-1) x_(4)) +...+ sqrt(sin^(-1) x_(28)) sqrt(cos^(-1) x_(1))` `x_(i) in [0,1] AA i = 1, 2, 3, ..,...

Let `x = tan theta`, where `theta in (-pi//2, pi//2)` `:. Tan^(-1). (3 x -x^(3))/(1 - 3x^(2)) = tan^(-1).(3 tan theta - tan^(3) theta)/(1 -3 tan^(2) theta)` `= tan^(-1)(tan 3...

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 - B Let `x = sin theta and sqrtx = sin phi, " where " x in [0, 1]` `rArr theta, phi in [0, pi//2]` `rArr theta - phi...

Correct Answer - i) `pi/3`, ii) `pi/6`, iii) `pi/3`, iv) `pi/4`, v) `pi/6`, vi) `pi/6`, vii) `pi/4`.

Correct Answer - i) `-pi/4`, ii) `(5pi)/6`, iii) `-pi/3`, iv) `(2pi)/3`, v) `-pi/4`, vi) `(2pi)/3`