First slide

Multiple and sub- multiple Angles

Question

Let $f (θ) = \frac{\cot θ}{1 - \tan θ} + \frac{\tan θ}{1 - \cot θ}, π < θ < \frac{3 π}{4}$ , then $f (\frac{9 π}{8})$ is equal to

Moderate

A

$1 + \sqrt{2}$
B

$1 + 2 \sqrt{2}$
C

$\sqrt{2} - 1$
D

$2 \sqrt{2} - 1$

Solution

$f (θ) = \frac{1}{\tan θ} \frac{1}{(1 - \tan θ)} + \frac{\tan θ \cdot \tan θ}{\tan θ - 1}$
$= \frac{1 - \tan^{3} θ}{\tan θ (1 - \tan θ)} = \frac{1 + \tan θ + \tan^{2} θ}{\tan θ}$
$= 1 + \frac{2}{\sin 2 θ}$
$∴ f (\frac{9 π}{8}) = 1 + \frac{2}{\sin (\frac{9 π}{4})} = 1 + 2 \sqrt{2}$

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