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Algebra of Derivatives

if x ϵ0,1, ...

Question

if $x ϵ (0, 1)$ , then the value of $t a n^{- 1} (\frac{1 - x^{2}}{2 x}) + c o s^{- 1} (\frac{1 - x^{2}}{1 + x^{2}})$ is equal to

A

- \frac{π}{2}

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B

z e r o

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C

\frac{π}{2}

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D

π

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Solution

The correct option is C $\frac{π}{2}$
Let $x = t a n A$
Then the above expression reduces to
$t a n^{- 1} (c o t 2 A) + c o s^{- 1} (c o s 2 A)$
$= \frac{π}{2} - c o t^{- 1} (c o t 2 A) + 2 A$
$= \frac{π}{2} - 2 A + 2 A$
$= \frac{π}{2}$

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Q.
(a) Prove that

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{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}

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