Is it possible to prove the following inequalities to be true?

I'm trying to prove the following inequalities are true, but they're a little too complex, is it even possible to prove that they are true? I suspect they are, but unless there is some simple method of proving it, I don't know if it's possible. Where $$0\leq \theta_{i,L}, \theta_{i,R} \leq \frac{\pi}{4}$$ and $$0 \leq \beta < \alpha <1$$. It's basically trying to show that if you decrease $$\theta_{i,R}$$ within the allowable range then the results of all 3 equations will decrease as well. In fact the inequalities may not be necessary, the main point is showing that the original equations (on the right side of each inequality) increase as $$\theta_{i,R}$$ increases, and decrease as $$\theta_{i,R}$$ decreases.

• If you just want to show that a certain function is monotonous, why don't you just look at its derivatives? – weee Nov 14 '18 at 21:40
• @weee Would this still work just by taking the partial derivative of each equation with respect to $\theta_{i,R}$ and showing that the results are strictly positive on the given interval? – RoryHector Nov 14 '18 at 21:57
• shure, if you are just interested how the term behaves when you change $\theta_{i,R}$ you can consider the other variables constant. – weee Nov 15 '18 at 9:28