Question

If $\tan \theta =\frac{5}{12}$, find the value of $(\sin \theta +\cos \theta )\sec \theta$.

Answer 1

$\tan \theta =\frac{5}{12}$

$\sec \theta =\sqrt{1+{\tan }^2\theta }=\sqrt{1+\frac{25}{144}}=\frac{13}{12}$

$\cos \theta =\frac{1}{\sec \theta }=\frac{12}{13}$

And, $\sin \theta =\sqrt{1-{\cos }^2\theta }=\sqrt{1-\frac{144}{169}}=\frac{5}{13}$

Now,

$(\sin \theta +\cos \theta )\sec \theta =(\frac{5}{13}+\frac{12}{13})\frac{13}{12}$

$=\frac{17}{13}\times \frac{13}{12}=\frac{17}{12}$