My Head of Department recently attended an IB Course in whch the Course Leader asked the group of teachers to solve this problem. Most teachers used either the double angle formula for tan or sine which is probably the most intuitive way. The IB Course Leader said afterwards that the reason that this is such a rich problem for students is that it can be solved using a number of different approaches. For example using vectors, matrices or complex numbers.
Throughout history, mathematicians have benefited hugely from knowing that a solution to a problem exists. Indeed, it´s much easier to persist with something if you know that it can be solved.
I personally would never have intuitively looked at the problem above and immediately started to solve it using complex numbers. I only proved it using complex numbers after having been told that this is one way to do it. I guess the main point is that it made me reflect on the assumptions I make about particular problems and to try not dive straight into solving a problem without giving it a little thought first. You never know what types of interesting methods you could use with a little bit of extra thought.
Here´s the proof I used for interest.