Question

If $\tan \alpha =\sqrt{\alpha }$, where $\alpha$ is a rational number which is not a perfect square then which of the following is a rational number?

• A. $\sin 2\alpha$
• B. $\tan 2\alpha$
• C. $\cos 2\alpha$
• D. none of these

Answer 1

Answer: (C)

$\cos 2\alpha$

$\tan \alpha =\sqrt{\alpha }$ where $\alpha$ is not a perfect square.
$\sin 2\alpha =\frac{2\tan \alpha }{1+{\tan }^2\alpha }=\frac{2\sqrt{\alpha }}{1+\alpha }$
$\tan 2\alpha =\frac{2\tan \alpha }{1-{\tan }^2\alpha }=\frac{2\sqrt{\alpha }}{1-\alpha }$
$\cos 2\alpha =\frac{1-{\tan }^2\alpha }{1+{\tan }^2\alpha }=\frac{1-\alpha }{1+\alpha }$
Therefore $\cos 2\alpha$ is a rational number.
Ans: C