Question

If $\sec \theta +\tan \theta =p$, find the value of $\csc \theta$

Answer 1

As $\sec 2\theta -\tan 2\theta =1$

$(\sec \theta -\tan \theta )P=1$

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

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

eqn. (1) + (2)

$\sec \theta =\frac{1}{2}(p+\frac{1}{p})$ ...(3)

eqn. (1) - (2)

$\tan \theta =-\frac{1}{2}(\frac{1}{p}-p)$ ...(4)

eqn. (3) $÷$ (4)

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