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To solve the trigonometric identity, visualizing the angles $\theta$ and $2\theta$ through a sketch can be beneficial. This approach helps in understanding the relationships between the angles and their corresponding sine and cosine values. Identifying key points on the unit circle may also clarify the problem. Engaging with visual aids often simplifies complex trigonometric concepts. Utilizing sketches can enhance comprehension and facilitate solving the identity.
For the sake of argument, lets assume this is, in fact, a right triangle with a rectangle inscribed in it.
Here is what looks like a very elegant solution:
Triangles A and B are similar
a/4=2/b
ab=8
But what is happening in that second line? Is there a property of similar triangles I'm forgetting?