Question

If $\sin \theta +\cos \theta =x$ and $\sec \theta +\text{cosec}\theta =y$, show that $(x^2-1)y=2x$.

Answer 1

Given, $x=\sin \theta +\cos \theta$ and $y=\sec \theta +\text{cosec}\theta$

Therefore, $x^2=1+2\sin \theta .\cos \theta$

$\Rightarrow x^2-1=2\sin \theta .\cos \theta$

Therefore, $(x^2-1)y=2\sin \theta .\cos \theta (\frac{1}{\cos \theta }+\frac{1}{\sin \theta })$

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