First slide

Multiple and sub- multiple Angles

Question

Let $f (θ) = \frac{\tan^{3} θ}{1 + \tan^{2} θ} - \frac{\cot^{3} θ}{1 + \cot^{2} θ}, 0 < θ < \frac{π}{4}$ Then $f (θ)$ is equal to

Moderate

A

$\tan θ + \cot θ$
B

$2 \sin (2 θ)$
C

$- 2 \cot (2 θ)$
D

0

Solution

$\begin{array}{l} f (θ) = \frac{\sin^{3} θ}{\cos θ} - \frac{\cos^{3} θ}{\sin θ} \\ = \frac{\sin^{4} θ - \cos^{4} θ}{\cos θ \sin θ} - \frac{- (\cos^{2} θ - \sin^{2} θ) (1)}{\cos θ \sin θ} \\ = - 2 \cot (2 θ) \end{array}$

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