First slide

Properties of ITF

Question

If $f 0 \leq x \leq 1,$ 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

none of these

Solution

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

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