# Prove $\sin(x)^3+\cos(x)^3 \leq 1$

$$\cos(x)\sin(x)\leq \frac{1}{2}$$

$$\cos(x)+\sin(x)\leq \sqrt{2}$$

How can I use these to prove inequality $$\sin(x)^3+\cos(x)^3 \leq 1$$? I tried to use binomial expansion, but this doesn't get me anywhere.

• Are you sure with your inequalities? Jan 2, 2020 at 10:48
• Please read this tutorial on how to typeset mathematics on this site. Jan 2, 2020 at 10:50

$$\sin^3x\le \sin^2x\tag1\label{1}$$ $$\cos^3x\le \cos^2x\tag2\label{2}$$ Add \eqref{1} and \eqref{2} to get the result.

• Is that because |cos(x)|≤1? (And same for sine) Jan 2, 2020 at 10:48
• @aradarbel10, Yes Jan 2, 2020 at 10:48
• @aradarbel10 ... and $\sin^2+\cos^2=1$ Jan 2, 2020 at 10:49
• You don't need those absolute values, do you ?
– user65203
Jan 2, 2020 at 10:49
• @Yves Daoust, No, I don't , but it shows that something stronger is true. Jan 2, 2020 at 10:50

This is not an answer but a comment which graph cannot be edited in the comments section.

No need to add something to Martund's smart proof (which can be generalized).

A stronger inequality : $$-1\leq\cos^n(x)+\sin^n(x)\leq 1\qquad n\geq 2$$

• Hi Jean ! Happy New Year ! Jan 2, 2020 at 12:08

Use $$\sin^2(x)+\cos^2(x)=1$$ to show $$\sin^3(x)+\cos^3(x) = (\sin(x) + \cos(x))(1-\sin(x)\cos(x))$$ and then apply the (corrected) inequalities.