Question

Prove the following identities (1-17)

$\cos \mathrm{\theta }\left(\tan \mathrm{\theta }+2\right)\left(2\tan \mathrm{\theta }+1\right)=2\sec \mathrm{\theta }+5\sin \mathrm{\theta }$

Answer 1

$$\begin{gathered} \mathrm{LHS}=\cos \theta \left(\tan \theta +2\right)\left(2\tan \theta +1\right) \\ =\cos \theta \left(2{\tan }^2\theta +5\tan \theta +2\right) \\ =\cos \theta \left(\frac{2{\sin }^2\theta }{{\cos }^2\theta }+\frac{5\sin \theta }{\cos \theta }+2\right) \\ =\frac{2{\sin }^2\theta +5\sin \theta \cos \theta +2{\cos }^2\theta }{\cos \theta } \\ =\frac{2+5\sin \theta \cos \theta }{\cos \theta } \\ =2sec\theta +5\sin \theta \\ =\mathrm{RHS} \\ \mathrm{Hence}\mathrm{proved}. \end{gathered}$$