How $\frac{\cos \alpha_1}{\sin \alpha}+\frac{\cos \beta_1}{\sin \beta}+\frac{\cos \gamma_1}{\sin \gamma}\leq\cot \alpha+\cot \beta+\cot \gamma$

Question

Let are any two triangles with angles $\alpha, \beta, \gamma$ and $\alpha_1, \beta_1, \gamma_1$. How prove that $$\frac{\cos \alpha_1}{\sin \alpha} + \frac{\cos \beta_1}{\sin \beta}+ \frac{\cos \gamma_1}{ \sin \gamma} \leq \cot \alpha + \cot \beta + \cot \gamma?$$

Answer 1

Hint:

$$ \frac{ \cos \alpha - \cos ( \alpha + \beta - \theta) } { \sin \alpha} + \frac{ \cos \beta - \cos \theta } { \sin \beta } = 2 \csc \alpha \csc \beta \sin ( \alpha + \beta ) \sin^2 ( \frac{ \beta - x } { 2} ) $$

Note that the RHS is always positive for $ 0 \leq \alpha, \beta \leq \pi $.

Apply this inequality twice, and you are done.