The Dirichlet integral… with the integrand squared!

Before looking at the solution, give the problem a go!

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A famous result named after Dirichlet is as follows:

The question is, how does our result transform if we square the integrand? Let us give the problem a go. Our integral is the following:

Let us remind ourselves of the trigonometric identity cos 2x = 1–2sin²x. We can apply it here to yield:

However, we know that this is the same as the following:

The viewer may already know where this is going after writing it in this form, and that is in fact, a double integral.

This seems much more familiar. We now swap the order of the integrals and get a brilliant result; Dirichlet’s integral:

Which then simplifies to:

Finally giving us the result:

Which is the exact same as Dirichlet’s integral! A beautiful result for a beautiful integral.