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  4. If sec θ=m and tan θ=n , then (1/m)[(m+n)+(1/(m+n))] is :

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Q. If $\sec \theta=m$ and $\tan \theta=n$ , then $\frac{1}{m}\left[(m+n)+\frac{1}{(m+n)}\right] $ is :

KCETKCET 2006Trigonometric Functions

Solution:

Given that $\sec\theta =m \tan\theta = n $
$\therefore \frac{1}{m} \left[\left(m+n\right) + \frac{1}{\left(m+n\right)}\right] $
$ = \frac{1}{\sec\theta} \left[\sec\theta + \tan\theta + \frac{1}{\sec\theta +\tan \theta}\right]$
$ = \frac{\left[\sec^{2} \theta + \tan^{2} \theta +2 \sec\theta \tan\theta+1\right]}{\sec\theta\left(\sec\theta +\tan\theta\right)} $
$= \frac{2 \sec^{2} \theta + 2 \sec\theta \tan \theta}{\sec\theta \left(\sec\theta + \tan \theta\right)} $
$ = \frac{2 \sec\theta \left(\sec \theta + \tan \theta\right)}{\sec \theta \left(\sec\theta + \tan \theta\right)} $
$ = 2 $