Trigo-Algebraic Inequality!

We can write the given expression as, $f(x)=x^2.(2.tan(x))/(1+tan^2(x/2)) + x.(1-tan^2(x/2))/(1+tan^2(x/2)) + x^2 + 1/2$

Let $m=tan(x/2)$

Therefore, $f(x)=x^2.(2.m/(1+m^2)) +x.(1-m^2)/(1+m^2) + x^2 + 1/2$

Simplifying we get. $f(x)=x^2.(1+m)^2 + x.(1-m^2) + (m^2+1/2)$

Discriminant of this quadratic equation in x, $D=(1-m^2)^2 -4.(1+m)^2.(m^2+1/2)$

Simplifying D we get,

$D= -(m+1)^2.(3.m^2+2.m +1)$

$3.m^2+2.m+1>0$ as discriminant of this equation is less than 0 and coefficient of m^2 is positive

Therefore, $D<0$ and coefficient of x^2 is positive.

Hence, $f(x)>0$ .