Question 48, the range of negative values for $a$ that make the inequality $\sin ^{2} x+a \cos x+a^{2} \geq 1+\cos x$ hold for all $x \in R$ is $\qquad$ -
Question 48, Solution: Notice that
$$
\begin{array}{l}
\sin ^{2} x+a \cos x+a^{2} \geq 1+\cos x \\
\Leftrightarrow \cos ^{2} x+(1-a) \cos x-a^{2} \leq 0
\end{array}
$$

Let $f(x)=x^{2}+(1-a) x-a^{2}$, then $f(x)$ is monotonically decreasing on $\left(-\infty, \frac{a-1}{2}\right)$ and monotonically increasing on $\left(\frac{a-1}{2},+\infty\right)$. Therefore, (*) is equivalent to: $f(1) \leq 0 \Leftrightarrow 1+(1-a)-a^{2} \leq 0 \Leftrightarrow(a+2)(a-1) \geq 0$. Hence, the range of negative values for $a$ is $(-\infty,-2]$.