First slide

Multiple and sub- multiple Angles

Question

Let $f : (- 1, 1) \to R$ be such that $f (\cos 4 θ) = \frac{2}{2 - \sec^{2} θ}$ for $θ \in (0, \frac{π}{4}) \cup (\frac{π}{4}, \frac{π}{2})$ . Then the value(s) of $f (\frac{1}{3})$ is(are)

Moderate

A

$1 - \sqrt{\frac{3}{2}}$
B

$1 + \sqrt{\frac{3}{2}}$
C

$1 - \sqrt{\frac{2}{3}}$
D

$1 + \sqrt{\frac{2}{3}}$

Solution

For $θ \in (0, \frac{π}{4}) \cup (\frac{π}{4}, \frac{π}{2})$
Let $\cos 4 θ = \frac{1}{3}$
$\Rightarrow \cos 2 θ = \pm \sqrt{\frac{1 + \cos 4 θ}{2}} = \pm \sqrt{\frac{2}{3}}$
$f (\frac{1}{3}) = \frac{2}{2 - \sec^{2} θ} = \frac{2 \cos^{2} θ}{2 \cos^{2} θ - 1} = 1 + \frac{1}{\cos 2 θ}$
$\Rightarrow f (\frac{1}{3}) = 1 - \sqrt{\frac{3}{2}} or 1 + \sqrt{\frac{3}{2}}$

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