tan^(- 1)((sqrt(1+x^2)+sqrt(1-x^2))/(sqrt(1+x^2)-sqrt(1-x^2)))=pi/4+1/2 cos^(- 1)x^2

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{cos}^{- 1} (x^{2} + \frac{1}{x^{2}} - 1) + {sin}^{- 1} (x^{2} - \frac{1}{x^{2}}) + {tan}^{- 1} (x^{2})

is equal to

(where x \in R - {0})

Q. Solve:

{cos}^{- 1} (\frac{x^{2} - 1}{x^{2} + 1}) + {tan}^{- 1} \frac{2 x}{x^{2} - 1} = \frac{2 π}{3}

Q.
(a) Prove that

sin [{t a n}^{- 1} \frac{1 - x^{2}}{2 x} + {cos}^{- 1} \frac{1 - x^{2}}{1 + x^{2}}] = 1

(b) If

{sin}^{- 1} (x - \frac{x^{2}}{2} + \frac{x^{3}}{4} - . . . . .) + {cos}^{- 1} (x^{2} - \frac{x^{4}}{2} + \frac{x^{6}}{4} - . . . . . . .) = \frac{π}{2}

for

0 < | x | < \sqrt{2}

, then x equals

Q. Solve the equation:

{cos}^{- 1} (\frac{x^{2} - 1}{x^{2} + 1}) + {sin}^{- 1} (\frac{2 x}{x^{2} + 1}) + {tan}^{- 1} (\frac{2 x}{x^{2} - 1}) = \frac{2 π}{3}

Prove that $t a n^{- 1} (\frac{\sqrt{1 + x^{2}} + \sqrt{1 - x^{2}}}{\sqrt{1 + x^{2}} - \sqrt{1 - x^{2}}}) = \frac{π}{4} + \frac{1}{2} c o s^{- 1} x^{2} .$

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