Question

Prove that $\frac{1-\cos 2\theta +\sin 2\theta }{1+\cos 2\theta +\sin 2\theta }=\tan \theta$

Answer 1

According to question,

$\frac{1-\cos 2\theta +\sin 2\theta }{1+\cos 2\theta +\sin 2\theta }=\tan \theta$

Solving LHS,

$⟹\frac{2{\sin }^2\theta +2\cos \theta sin\theta }{2{\cos }^2\theta +2\sin \theta .\cos \theta }$

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

$⟹\frac{\sin \theta }{\cos \theta }$

$⟹\tan \theta$=RHS

Hence proved