Assuming $x \in (0, \frac{\pi}{4})$ Write down a proof that $F(x) > 0 $ for all $x$'s where $F(x) = \tan(\sin x)-\sin(\tan x)$.
All I came up with was using Jensen inequity for 1st derivative using $F(x) = \log(\cos x)$ and $\alpha_1 =\alpha_2 = \alpha_3 = \frac{1}{3}$ and gaining $3\log(\cos(2\sin x\cdot \tan x)) \geq 2\log(\cos(\sin x)) + \log(\cos(\tan x))$ but what I need to gain is this inequality: $3\log(\cos x) \geq \leq 2\log(\cos(\sin x)) + \log(\cos(\tan x))$ Any hint would be helpful:p
