Question

If $\sec \theta +\tan \theta =p$, obtain the value of $\sec \theta$, $\tan \theta$, $\sin \theta$ in terms of $p$.

Answer 1

As we know ${\sec }^2\theta -{\tan }^2\theta =1$,

then,$\left(\sec \theta -\tan \theta \right)\left(\sec \theta +\tan \theta \right)=1$

$\sec \theta -\tan \theta =\frac{1}{\sec \theta +\tan \theta }$

$\sec \theta -\tan \theta =\frac{1}{p}$........................(1)

$\sec \theta +\tan \theta =p$.......................................(2)

Add equation (1) and (2),

$\sec \theta =\frac{1}{2}\left(\frac{p^2+1}{p}\right)$

Subtract equation (1) and (2),

$\tan \theta =\frac{1}{2}\left(\frac{p^2-1}{p}\right)$

From the values of $\sec \theta$ and $\tan \theta$,

$\sin \theta =\frac{p^2-1}{p^2+1}$