The sum of possible values of x for

tan^$$-$$1(x + 1) + cot^$$-$$1$$\left( {{1 \over {x - 1}}} \right)$$ = tan^$$-$$1$$\left( {{8 \over {31}}} \right)$$ is :

If cot^$$-$$1($$\alpha$$) = cot^$$-$$1 2 + cot^$$-$$1 8 + cot^$$-$$1 18 + cot^$$-$$1 32 + ...... upto 100 terms, then $$\alpha$$ is :

Given that the inverse trigonometric functions take principal values only. Then, the number of real values of x which satisfy

$${\sin ^{ - 1}}\left( {{{3x} \over 5}} \right) + {\sin ^{ - 1}}\left( {{{4x} \over 5}} \right) = {\sin ^{ - 1}}x$$ is equal to :

If 0 < a, b < 1, and tan^$$-$$1a + tan^$$-$$1b = $${\pi \over 4}$$, then the value of

$$(a + b) - \left( {{{{a^2} + {b^2}} \over 2}} \right) + \left( {{{{a^3} + {b^3}} \over 3}} \right) - \left( {{{{a^4} + {b^4}} \over 4}} \right) + .....$$ is :