First slide

Properties of ITF

Question

If $\sin^{- 1} \sqrt{x^{2} + 2 x + 1} + \sec^{- 1} \sqrt{x^{2} + 2 x + 1} = \frac{π}{2}, x \neq 0$ then the value of $2 \sec^{- 1} \frac{x}{2} + \sin^{- 1} \frac{x}{2}$ is equal to

Moderate

A

$- \frac{π}{2}$ only
B

$\{- \frac{3 π}{2}, \frac{π}{2}\}$
C

$\frac{3 π}{2}$ only
D

$- \frac{3 π}{2}$ only

Solution

We have,
$\begin{array}{l} \sin^{- 1} \sqrt{x^{2} + 2 x + 1} + \sec^{- 1} \sqrt{x^{2} + 2 x + 1} = \frac{π}{2} \\ \sin^{- 1} \sqrt{x^{2} + 2 x + 1} + \cos^{- 1} \frac{1}{\sqrt{x^{2} + 2 x + 1}} = \frac{π}{2} \\ \Rightarrow \sqrt{x^{2} + 2 x + 1} = \frac{1}{\sqrt{x^{2} + 2 x + 1}} \\ \Rightarrow x^{2} + 2 x + 1 = 1 \Rightarrow x = 0, - 2 . \end{array}$
For $x = 0,$ we find that $\sec^{- 1} \frac{x}{2}$ is not defined.
For $x = - 2,$ we have
$\begin{array}{l} 2 \sec^{- 1} \frac{x}{2} + \sin^{- 1} \frac{x}{2} = 2 \sec^{- 1} (- 1) + \sin^{- 1} (- 1) \\ = 2 \times π - \frac{π}{2} = \frac{3 π}{2} \end{array}$

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