When integrating by trigonometric substitution that has $\sqrt{a^2-x^2}$ in the integrand, the general recommended approach is to make the substitution $x = a \sin\theta$, and this substitution is the one shown in countless place, with the accompanying proof.
However, I have never seen the substitution $x = a\cos\theta$, which seems odd, since it seems the proof would work equally well for cosine as it does for sine. Is there any particular reason this substitution seems to never be used, besides custom?
The two substitutions are virtually identical. A slight complication arises if you set $x=a\cos(\theta)$, for then $dx=-a\sin(\theta)d\theta$. Note that you pick up a minus sign that you don't pick up with $x=a\sin(\theta)$.