Question

Prove: $\frac{{(\sin \theta +\cos \theta )}^2-{(\sin \theta -\cos \theta )}^2}{{(\sin \theta +\cos \theta )}^2+{(\sin \theta -\cos \theta )}^2}=2\sin \theta \cos \theta$

Answer 1

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

$=\frac{4\sin \theta \cos \theta }{1+1}$ $(\because {\sin }^2\theta +{\cos }^2\theta =1)$

$=2\sin \theta \cos \theta =RHS$