Question

Prove that:
${\sin }^2{\theta }_{\cdot }\tan \theta +{\cos }^2{\theta }_{\cdot }\cot \theta +2\sin {\theta }_{\cdot }\cos \theta =\tan \theta +\cot \theta$

Answer 1

L.H.S

${\sin }^2{\theta }_{\cdot }\tan \theta +{\cos }^2{\theta }_{\cdot }\cot \theta +2\sin {\theta }_{\cdot }\cos \theta$

$\Rightarrow {\sin }^2\theta \frac{\sin \theta }{\cos \theta }+{\cos }^2\theta \frac{\cos \theta }{\sin \theta }+2\sin \theta \cos \theta$

$\Rightarrow \frac{{\sin }^3\theta }{\cos \theta }+\frac{{\cos }^3\theta }{\sin \theta }+2\sin \theta \cos \theta$

$\Rightarrow \frac{{\sin }^4\theta +{\cos }^4\theta }{\sin \theta \cos \theta }+2\sin \theta \cos \theta$

$\Rightarrow \frac{{\sin }^4\theta +{\cos }^4\theta +2{\sin }^2\theta {\cos }^2\theta }{\sin \theta \cos \theta }$

$\Rightarrow \frac{({\sin }^2\theta +{\cos }^2\theta {)}^2}{\sin \theta \cos \theta }$

$\Rightarrow \frac{(1{)}^2}{\sin \theta \cos \theta }$

$\Rightarrow \frac{1}{\sin \theta \cos \theta }$

$\Rightarrow \frac{{\sin }^2\theta +{\cos }^2\theta }{\sin \theta \cos \theta }$

$\Rightarrow \frac{\sin \theta }{\cos \theta }+\frac{\cos \theta }{\sin \theta }$

$\Rightarrow \tan \theta +\cot \theta$

R.H.S

Hence, proved.