Question

If $tan\theta =\sqrt{n}$ for some non-square natural number $n$, then $sec2\theta$ is :

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

Answer 1

Answer: (A)

$cos2\theta =\frac{1-ta{n}^2\theta }{1+ta{n}^2\theta }=\frac{1-n}{1+n}\Rightarrow sec2\theta =\frac{1+n}{1-n}$
Since, $n$ is non-square natural number $\therefore n\ne 1$
$\therefore sec2\theta =\frac{1+n}{1-n}=$ rational number