1. The number of real roots of the equation $1-\lg \sin x=\cos x$ is ( ).
(A) 0 .
(B) 1
(C) 2
(D) greater than 2
-1 (A).
Since for all $x$, $\sin x \leqslant 1$, it follows that $\lg \sin x \leqslant 0$ and $1-\lg \sin x \geqslant 1$. However, for any $x$, $\cos x \leqslant 1$, so for all $x$, $1-\lg \sin x \geqslant \cos x$, with equality holding if and only if $\sin x=\cos x=1$, which is impossible.