First slide

Properties of ITF

Question

If $\frac{1}{2} \leq x \leq 1$ then $\cos^{- 1} (\frac{x}{2} + \frac{1}{2} \sqrt{3 - 3 x^{2}}) + \cos^{- 1} x =$

Moderate

A

$\frac{π}{3}$
B

$\frac{π}{6}$
C

$\frac{π}{2}$
D

$\frac{π}{4}$

Solution

$\begin{array}{l} \cos^{- 1} x + \cos^{- 1} (\frac{x}{2} + \frac{1}{2} \sqrt{3 - 3 x^{2}}) \\ = \cos^{- 1} x + \cos^{- 1} (x . \frac{1}{2} + \frac{\sqrt{3}}{2} \sqrt{1 - x^{2}}) \\ = \cos^{- 1} x + \cos^{- 1} (\frac{1}{2}) - \cos^{- 1} x \\ = \cos^{- 1} (\frac{1}{2}) \\ = \frac{π}{3} \end{array}$

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