Question

$x=\sec \theta -\tan \theta ,y=cosec\theta +\cot \theta$
Prove: $xy+1=y-x$

Answer 1

Given,

$x=\sec \theta -\tan \theta ,y=cosec\theta +\cot \theta$.

Now, $xy+1$

$=(\sec \theta -\tan \theta )(cosec\theta +\cot \theta )$

$=\sec \theta .cosec\theta +\sec \theta .\cot \theta -\tan \theta .cosec\theta -1+1$

$=\frac{1}{\sin \theta .\cos \theta }+cosec\theta -\sec \theta$

$=\frac{{\sin }^2\theta +{\cos }^2\theta }{\sin \theta .\cos \theta }+cosec\theta -\sec \theta$ [ Since ${\sin }^2\theta +{\cos }^2\theta =1$]

$=\tan \theta +\cot \theta +cosec\theta -\sec \theta$.

$=(cose\theta +\cot \theta )-(\sec \theta -\tan \theta )$

$=y-x$.