$132 \mathrm{~B}$.
Since $0<\sin \alpha, \cos \alpha<1$, the function $f(x)=\sin ^{x} \alpha+\cos ^{x} \alpha$ is decreasing. Therefore, the original equation has no more than one root. It is also clear that $x=2$ is a root of the equation.
132 Given $0<\alpha<\frac{\pi}{2}$, the number of solutions to the equation $\sin ^{x} \alpha+\cos ^{x} \alpha=1$ is
A. 0
B. 1
C. 2
D. Greater than 2