Question

If $\tan \theta =\sqrt{n}$, where $n\in N\ge 2$, then $\sec 2\theta$ is always

• A. a rational number
• B. an irrational number
• C. a positive integer
• D. a negative integer

Answer 1

The correct option is A a rational number

$\tan \theta =\sqrt{n}$
$\Rightarrow \cos 2\theta =\frac{1-{\tan }^2\theta }{1+{\tan }^2\theta }\Rightarrow \cos 2\theta =\frac{1-n}{1+n}$
$\Rightarrow \sec 2\theta =\frac{1+n}{1-n}$
As $n\in N\ge 2$, so $\frac{1+n}{1-n}$ is a rational number.