Show that for all integers m and n, with m ≠ ±n,
∫_(-π)^π cos⁡(mθ)cos⁡(nθ)dθ=0.

In English that is "the integral from negative pi to pi of..."
If it helps, please note this is in the section involving integration by partial fractions and trig substitution.

Question

Show that for all integers m and n, with m ≠ ±n, 
∫_(-π)^π cos⁡(mθ)cos⁡(nθ)dθ=0.   

In English that is "the integral from negative pi to pi of..." 
If it helps, please note this is in the section involving integration by partial fractions and trig substitution.

anonymous · Answer

$$\cos p \ \cos q=\frac{1}{2} (\cos(p+q)+\cos(p-q))$$

anonymous · Answer

Is that just a trig identity that should be memorized? I do not see that in this section? What exactly am I supposed to do with that?

turingtest · Answer

just integrate what @mukushla  showed you and evaluate

turingtest · Answer

you can also make use of the fact that cosine is an even function, and for even functions$$\int_{-a}^af(x)dx=2\int_0^af(x)dx$$to save some trouble in the evaluation

anonymous · Answer

Thanks! Both of you!