Which of the following statements are always correct (where Q denotes the set of rationals)?

Moderate

A

$\cos 2 θ \in Q$ and $\sin 2 θ \in Q \Rightarrow \tan θ \in Q$ $and \cos θ \in Q \Rightarrow \tan 3 θ \in Q$
B

$\tan θ \in Q \Rightarrow \sin 2 θ, \cos 2 θ$ and $\tan 2 θ \in Q$ (if defined)
C

$\sin θ \in Q$ and $\cos θ \in Q \Rightarrow \tan 3 θ \in Q$ (if defined)
D

$\sin θ \in Q \Rightarrow \cos 3 θ \in Q$

Solution

a. $\tan θ = \frac{1 - \cos 2 θ}{\sin 2 θ}$ Hence, option (a) is correct.
b.
$\begin{aligned} \sin 2 θ = \frac{2 \tan θ}{1 + \tan^{2} θ} \\ \cos 2 θ = \frac{1 - \tan^{2} θ}{1 + \tan^{2} θ} \\ \tan 2 θ = \frac{2 \tan θ}{1 - \tan^{2} θ} \end{aligned}$
Hence, option (b) is correct.
c. $\tan 3 θ = \frac{\sin 3 θ}{\cos 3 θ}$ Hence, option (c) is correct.
d. $\sin θ = 1 / 3$ which is rational but $\cos 3 θ = \cos θ (4 \cos^{2} θ - 3)$ which is irrational.
Hence, option (d) is incorrect.

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