Question

If $tan\,\theta=\sqrt{n}$ for some non-square natural number $n$, then $sec\, 2\theta$ is :

• A. a rational number
• B. an irrational number
• C. a positive number
• D. none of the above

Answer 1

Answer: (A)

$cos\,2\theta=\frac{1-tan^{2}\,\theta}{1+tan^{2}\,\theta}=\frac{1-n}{1+n} \Rightarrow sec\,2\theta=\frac{1+n}{1-n}$
Since, $n$ is non-square natural number $\therefore n\ne1$
$\therefore sec\,2\theta =\frac{1+n}{1-n}=$ rational number