Question

if sec theta +tan theta =p .prove that tan theta =1/2 (p-1/p)​

Answer 1

Answer:

Given:

• $\sec\theta + \tan\theta = p$

To prove :

• $\tan\theta = \dfrac{1}{2} \left(p -\dfrac{1}{p}\right)$

Solution:

$\sec\theta + \tan\theta = p$⠀⠀⠀⠀⠀⠀⠀⠀-(i)

$Using \ identity ,$

$\boxed{1 + \tan^2\theta = \sec^2\theta}$

$\implies \sec^2\theta - \tan^2\theta = 1$

$\implies (\sec\theta + \tan \theta )(\sec\theta - \tan \theta) = 1$

$From \ (i) ,$

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

$\implies \sec\theta - \tan\theta = \dfrac{1}{p}$⠀⠀⠀-(ii)

$Now \ (i) - (ii)$

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

$2\tan\theta = p - \dfrac{1}{p}$

$\boxed{\implies{\boxed{\tan\theta = \dfrac{1}{2} \left(p-\dfrac{1}{p}\right)}}}$