Question

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

• A. $\cos 2\theta ϵQ$ and $\sin 2\theta ϵQ\Rightarrow \tan \theta ϵQ$ ( if defined)
• B. $\tan \theta ϵQ\Rightarrow \sin 2\theta ,\cos \theta$ and $\tan 2\theta ϵQ$ ( if defined)
• C. $\sin \theta ϵQ$ and $\cos \theta ϵQ\Rightarrow \tan 3\theta ϵQ$ (if defined)
• D. $\sin \theta ϵQ\Rightarrow \cos 3\theta ϵQ$

Answer 1

Answer: (A), (B), (C)

The correct options are
A $\cos 2\theta ϵQ$ and $\sin 2\theta ϵQ\Rightarrow \tan \theta ϵQ$ ( if defined)
B $\tan \theta ϵQ\Rightarrow \sin 2\theta ,\cos \theta$ and $\tan 2\theta ϵQ$ ( if defined)
C $\sin \theta ϵQ$ and $\cos \theta ϵQ\Rightarrow \tan 3\theta ϵQ$ (if defined)

$\sin 2\theta =\frac{2\tan \theta }{1+{\tan }^2\theta }$
$\cos 2\theta =\frac{1-{\tan }^2\theta }{1+{\tan }^2\theta }$.
$\tan 2\theta =\frac{2\tan \theta }{1-{\tan }^2\theta }$
Hence if $\cos 2\theta ϵQ$ , $\sin 2\theta ϵQ$ and $\tan 2\theta ϵQ$ then $\tan \theta ϵQ$.

Now

$\cos 3\theta =4{\cos }^3\left(x\right)-3\cos \left(x\right)$

Hence if $\sin \theta ϵQ$ it $\cos 3\theta$ may or may not belong to $Q$.