Question

Prove the following identities (1-17)

$\left(\mathrm{cosec}\mathrm{\theta }-\sin \mathrm{\theta }\right)\left(\sec \mathrm{\theta }-\cos \mathrm{\theta }\right)\left(\tan \mathrm{\theta }+\cot \mathrm{\theta }\right)=1$

Answer 1

$$\begin{gathered} \mathrm{LHS}=\left(\mathrm{cosec}\mathit{}\theta -\sin \mathit{}\theta \right)\left(\sec \mathit{}\theta -\cos \theta \right)\left(\tan \theta +\cot \theta \right) \\ =\left(\frac{1}{\sin \mathit{}\theta }-\sin \mathit{}\theta \right)\left(\frac{1}{\cos \theta }-\cos \theta \right)\left(\frac{\sin \mathit{}\theta }{\cos \theta }+\frac{\cos \theta }{\sin \mathit{}\theta }\right) \\ =\left(\frac{1-{\sin }^2\theta }{\sin \mathit{}\theta }\right)\left(\frac{1-{\cos }^2\theta }{\cos \theta }\right)\left(\frac{{\sin }^2\mathit{}\theta +{\cos }^2\theta }{\cos \theta \sin \mathit{}\theta }\right) \\ =\left(\frac{{\cos }^2\theta }{\sin \mathit{}\theta }\right)\left(\frac{{\sin }^2\mathit{}\theta }{\cos \theta }\right)\left(\frac{1}{\cos \theta \sin \mathit{}\theta }\right) \\ =1 \\ =\mathrm{RHS} \\ \mathrm{Hence}\mathrm{proved}. \end{gathered}$$