Question

Prove the following identities (1-17)

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

Answer 1

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