Question

Prove that $1+{\sin }^2\theta +{\sin }^2\varphi >\sin \theta +\sin \varphi +\sin \theta \sin \varphi$

Answer 1

If $a=1,b=sin\theta ,c=sin\varphi$, then we have to prove
$a^2+b^2+c^2-ab-bc-ca>0$
$or\frac{1}{2}[{(a-b)}^2+(b-c{)}^2+(c-a{)}^2]>0$
Above is true.