Can I use a property of the dirac delta function to evaluate a product of 2 functions?

Question

$$\int_0^a \sin (\frac{n\pi x}{a}) \delta(x-a/2)\sin (\frac{m\pi x}{a})dx$$

For the integral above, I cannot combine the two sine functions easily. But can I do this:

$$=\sin (\frac{n\pi}{2})\sin (\frac{m\pi}{2})$$

? Basically I am treating both sines multiplied together as one function of $x$ and applying the property of dirac delta to this big function. I have another idea to, but not sure which of these two ideas is correct:

$$=\frac{1}{2}\int_0^a \delta(x-a/2) (\cos(\frac{n\pi x}{a}- \frac{m\pi x}{a})-\cos(\frac{n\pi x}{a} + \frac{m\pi x}{a}))dx$$

$$=\frac{1}{2} (\cos(\frac{n\pi }{2}- \frac{m\pi }{2})-\cos(\frac{n\pi }{2} + \frac{m\pi }{2}))$$

Answer 1

They're both entirely correct (as long as $0<a<1$), and they're the same answer.

Answer 2

Both work: the product of two functions is a function, so there's no difficulty in doing it the first way. And you will notice, by using the prosthaphaeresis formula the other way, that the answers are equal.