<s>[INST] Prove the following trignometric inequality for all real $x$ Prove the following trignometric inequality for all $x \in \Bbb R$
$$x^2 \sin(x) + x \cos(x) + x^2 + {\frac 12} >0$$
take $x$  in the form of radians.
This particular question is the seventh question of the  1995 Indian RMO.
 [/INST] Put $t=\tan(\frac{x}{2})$. Now substitute it in the original expression. We have $$ x^{2}\sin x + x\cos x + x^{2} + \frac{1}{2}$$ $$= (1+\sin x)x^{2} + x\cos x+\frac{1}{2}$$ $$=(1 + \frac{2t}{1+t^{2}})x^{2} + x(\frac{1-t^{2}}{1+t^{2}}) + \frac{1}{2}$$ $$= \frac{(1+t)^{2}}{1+t^{2}}[x^{2} - x\frac{1-t^{2}}{(1+t)^{2}} + \frac{(1-t^{2})^{2}}{4(1+t)^{4}} + \frac{1+t^{2}}{4}]$$ $$= \frac{(1+t)^{2}}{1+t^{2}}[x-\frac{1-t^{2}}{2(1+t)^{2}}]^{2} + \frac{(1+t)^{2}}{4} > 0. $$ Hope it helps.
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