Question

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

Answer 1

$sec\theta +tan\theta =p$ ... (1)

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

(1) + (2)

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

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

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

(1) - (2)

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

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

$\Rightarrow \frac{sin\theta }{cos\theta }=\frac{1}{2}(p-\frac{1}{p})$

$\Rightarrow sin\theta =\left[\frac{1}{2}(p+\frac{1}{p})\right]\cos \theta$

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

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