Question

If $cosec\theta -\cot \theta =x,$ then $cosec\theta +\cot \theta =$

Answer 1

$cosec\theta -\cot \theta =x$

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

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

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

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

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

$\frac{\sin \theta }{1+\cos \theta }=x$

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

Now

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

Hence $cosec\theta +\cot \theta =\frac{1}{x}$