First slide

Properties of ITF

Question

$\cos^{- 1} \{\frac{1}{2} x^{2} + \sqrt{1 - x^{2}} \sqrt{1 - \frac{x^{2}}{4}}\} = \cos^{- 1} \frac{x}{2} - \cos^{- 1} x$ holds for

Moderate

A

$x \in [- 1, 1]$
B

$x \in R$
C

$x \in [0, 1]$
D

$x \in [- 1, 0]$

Solution

We observe that
$\frac{x^{2}}{2} + \sqrt{1 - x^{2}} \sqrt{1 - \frac{x^{2}}{4}}$ is positive and defined for all $x \in [- 1, 1]$
$∴ \cos^{- 1} \{\frac{x^{2}}{2} + \sqrt{1 - x^{2}} \sqrt{1 - \frac{x^{2}}{4}}\} \in [0, \frac{π}{2}]$
Now, RHS of the given relation is defined for
$|\frac{x}{2}| \leq 1$ and $| x | \leq 1$ i,e. for $x \in [- 1, 1]$
Also, LHS $\geq 0$
$\Rightarrow \cos^{- 1} \frac{x}{2} - \cos^{- 1} x \geq 0 \Rightarrow \cos^{- 1} \frac{x}{2} \geq \cos^{- 1} x \Rightarrow x \in [0, 1]$
Hence, LHS = RHS for $x \in [0, 1] .$

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