# Trigonometric inequalities with substitution

Using inequality $$\tan \frac{x}{2} > \frac{x}{2}$$ prove that $$\sin x > x- \frac{x^3}{4}$$

I tried with substitution $$\tan \frac{x}{2} = t$$

$$\sin x = \frac{2t}{t^2+1}$$

$$t>\frac{x}{2}$$

$$2t>x$$

$$t^2+1>1+\frac{x^2}{4}$$

• NB both the original and derived identities are only valid for $x > 0$. – Travis Nov 13 '18 at 9:55