1. Given $\sin ^{2005} x+\cos ^{2005} x=1$. Then for any $k>0$, we must have ( ).
(A) $\sin ^{k} x+\cos ^{k} x=1$
(B) $\sin ^{k} x+\cos ^{k} x>1$
(C) $\sin ^{k} x+\cos ^{k} x<1$
(D) The value of $\sin ^{k} x+\cos ^{k} x$ is uncertain
-.1.A.
Since $\sin ^{200} x+\cos ^{2008} x \leqslant \sin ^{2} x+\cos ^{2} x=1$, equality holds if and only if $\sin x$ and $\cos x$ take 0 and 1, respectively.
At this time, for any $k>0$, it must be that $\sin ^{k} x+\cos ^{k} x=1$.