Dear Uncle Colin

I’m stuck on a trigonometry proof: I need to show that $\cosec(x) – \sin(x) \ge 0$ for $0 < x < \pi$. How would you go about it?

– Coming Out Short of Expected Conclusion

Hi, COSEC, and thank you for your message! As is so often the case, there are several ways to approach this.

The most obvious one

The first approach I would try would be to turn the left hand side into a single fraction: $\frac{1}{\sin(x)} – \sin(x) \equiv \frac{1 – \sin^2(x)}{\sin(x)}$.

The top of that is $\cos^2(x)$, so you have $\frac{\cos^2(x)}{\sin(x)}$.

In the specified region,…

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