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I have the derivative:

$$- \frac1{x^2} + 1 + \frac{\cos^2(x)}{\sin^2(x)}$$

and am supposed to show that this is positive for all $x \in (n\pi, (n+1)\pi)$.

How exactly am I supposed to do that? I'm thinking there's a trick I'm supposed to use, but can't figure it out.

AlexR
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Mane
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  • Welcome to Math.SE! Have a look at this help page to see how to use MathJax for mathematical expressions. Note that the trigonometric part of your expression is $\cot^2(x)$ so you need to show that $$\cot^2(x) > \frac1{x^2} - 1$$ or that $$\frac1{\sin^2(x)} > \frac1{x^2} \Leftrightarrow \sin^2 x < x^2$$ in the given intervals. – AlexR Sep 18 '14 at 12:19

1 Answers1

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Hint: $1 + \frac{\cos^2 x}{\sin^2 x} = \frac{1}{\sin^2 x}$, so all you need to do is compare $x^2$ with $\sin^2 x$.

Clive Newstead
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