I asked this question without any limitation on methods that might be used. I believe it's turned out to be interesting to see a variety of different approaches. It turns out that the aim of the exercise that my students were given was to provide an example of the application of the Mean Value Theorem. I really like the MVT solution as it's a nice little application showing how powerful elementary calculus can be. None the less, I've enjoyed following other lines of thought.
I'm ticking Christian Blatter's solution as it's a particularly nice application of precalculus knowledge. It would be a good challenge question for brighter maths students in the final year of high school after introducing the various trigonometric formulae. It's a neat application of converting a difference to a product.
Finally, it's a good example of the contrast between pre and post calculus. Thanks to everyone who has contributed to this page.