First slide

Properties of ITF

Question

If $- 1 \leq x \leq 0,$ then $\cos^{- 1} (2 x^{2} - 1)$ equals

Moderate

A

$2 \cos^{- 1} x$
B

$π - 2 \cos^{- 1} x$
C

$2 π - 2 \cos^{- 1} x$
D

$- 2 \cos^{- 1} x$

Solution

Let $\cos^{- 1} x = θ .$ Then, $x = \cos θ$
Also,
$- 1 \leq x \leq 0 \Rightarrow - 1 \leq \cos θ \leq 0 \Rightarrow \frac{π}{2} \leq θ \leq π$
Now,
$\begin{array}{l} \cos^{- 1} (2 x^{2} - 1) = \cos^{- 1} (\cos 2 θ) = \cos^{- 1} (2 π - 2 θ) \\ = 2 π - 2 θ [∵ \frac{π}{2} \leq θ \leq π \Rightarrow π \leq 2 θ \leq 2 π] \\ \Rightarrow 2 π - 2 \cos^{- 1} x \end{array}$

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