 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2015-resit/12/a/ii/solution

m, n

, the product-to-sum identity gives

sin (n x) sin (m x) = \frac{1}{2} [cos ((n - m) x) - cos ((n + m) x)] .

Both cosine integrals vanish over

[- π, π]

, so

⟨ f_{n}, f_{m} ⟩ = 0 (m \neq = n) .

If one index is zero, the corresponding function is identically zero and the same answer follows immediately.

 New to topics? Read the docs here!