Question

If $x=\sec \theta -\tan \theta$ and $y=\text{cosec}\theta +\cot \theta$, then

• A. $xy+1=x-y$
• B. $xy-1=y-x$
• C. $xy+1=y+x$
• D. $xy+1=y-x$

Answer 1

The correct option is D $xy+1=y-x$

Given $x=\sec \theta -\tan \theta =\frac{1-\sin \theta }{\cos \theta }$

$y=\text{cosec}\theta +\cot \theta =\frac{1+\cos \theta }{\sin \theta }$

$xy+1=\left(\frac{1-\sin \theta }{\cos \theta }\right)\left(\frac{1+\cos \theta }{\sin \theta }\right)+1=\frac{1-\sin \theta +\cos \theta }{\sin \theta \cos \theta }$

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

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

$=(\text{cosec}\theta +\cot \theta )-(\sec \theta -\tan \theta )=y-x$

$\therefore xy+1=y-x$