Question

Evaluate,

$(\csc \theta -\sin \theta )(\sec \theta -\cos \theta )(\tan \theta +\cot \theta )=$

Answer 1

$\left(cosec\theta -\sin \theta \right)\left(\sec \theta -\cos \theta \right)\left(\tan \theta +\cot \theta \right)$

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

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

$=\frac{{\cos }^2\theta {\cos }^2\theta }{\sin \theta \cos \theta }\times \frac{1}{\sin \theta \cos \theta }$ $\left({\sin }^2\theta +{\cos }^2\theta =1\right)$

$=\frac{{\sin }^2\theta {\cos }^2\theta }{{\sin }^2\theta {\cos }^2\theta }$

$=1$