148 The number of solutions to the equation $\sin \left(x^{\sin x}\right)=\cos \left(x^{\cos x}\right)$ in the closed interval $\left[\frac{\pi}{4}, \frac{\pi}{2}\right]$ is
A. 0
B. 2
C. A finite number, more than 2
D. An infinite number
148 A. Clearly, the original equation is equivalent to $x^{\sin x}+x^{\cos x}=\frac{\pi}{2}\left(\frac{\pi}{4} \leqslant x \leqslant \frac{\pi}{2}\right)$. When $\frac{\pi}{4} \leqslant x \leqslant 1$, $x^{\sin x}, x^{\cos x}>x$, so $x^{\sin x}+x^{\cos x}>2 x \geqslant \frac{\pi}{2}$; when $1 \leqslant x \leqslant \frac{\pi}{2}$, $x^{\sin x} \geqslant 1, x^{\cos x} \geqslant 1$, so $x^{\sin x}+x^{\cos x} \geqslant 2>\frac{\pi}{2}$. Therefore, in the closed interval $\left[\frac{\pi}{4}, \frac{\pi}{2}\right]$, $x^{\sin x}+x^{\cos x}>\frac{\pi}{2}$, hence the original equation has no solution.