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

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

Solution:

$\sec \theta=m, \tan \theta=n$
$\frac{1}{m}\left[(m+n)+\frac{1}{m+n}\right]$
$=\frac{1}{\sec \theta}\left[\sec \theta+\tan \theta+\frac{1}{\sec \theta+\tan \theta}\right]$
$=\frac{1}{\sec \theta}[\sec \theta+\tan \theta+\sec \theta-\tan \theta]$
$=\frac{2 \sec \theta}{\sec \theta}=2$