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I already know:

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

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

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.

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

3 Answers 3

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

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

enter image description here

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  • $\begingroup$ Hi Jean ! Happy New Year ! $\endgroup$ Jan 2, 2020 at 12:08
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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.

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