Question

If $\sec \theta +\tan \theta =p$ then find the value of $\cot \theta$

Answer 1

given, $\sec \theta +\tan \theta =p$ .......$\left(1\right)$

$\Rightarrow (\sec \theta +\tan \theta )(\sec \theta -\tan \theta )=p(\sec \theta -\tan \theta )$

$\Rightarrow {\sec }^2\theta -{\tan }^2\theta =p(\sec \theta -\tan \theta )$

$\Rightarrow 1=p(\sec \theta -\tan \theta )$

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

$\left(2\right)-\left(1\right)$ we get

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

$\Rightarrow 2\tan \theta =\frac{p^2-1}{p}$

$\Rightarrow \tan \theta =\frac{p^2-1}{2p}$

$\Rightarrow \cot \theta =\frac{2p}{p^2-1}$