Question

If $\left[sin\theta \right]+cos\left[\theta \right]=p$and $sec\left[\theta \right]+cosec\left[\theta \right]=q$ then show that,$q[\left(p^2-1\right)=2p]$

Answer 1

Given that, $\sin \theta +\cos \theta =p$ and $\sec \theta +\cos ec\theta =q$

Then prove that $q(p^2-1)=2p$

$L.H.S.$

$q(p^2-1)$

$=(\sec \theta +\cos ec\theta )[{(\sin \theta +\cos \theta )}^2-1]$

$=(\sec \theta +\cos ec\theta )[{\sin }^2\theta +{\cos }^2\theta +2\sin \theta \cos \theta -1]$

$=(\sec \theta +\cos ec\theta )[1+2\sin \theta \cos \theta -1]$

$=(\sec \theta +\cos ec\theta )\left(2\sin \theta \cos \theta \right)$

$=2\sin \theta \cos \theta (\frac{1}{\cos \theta }+\frac{1}{\sin \theta })$

$=2\sin \theta \cos \theta \left(\frac{\sin \theta +\cos \theta }{\sin \theta \cos \theta }\right)$

$=2(\sin \theta +\cos \theta )$

$=2p$

$R.H.S.$

Hence proved.