Let $\csc A=\frac{\sqrt{5}}{2}$ with $A$ in $Q I$ and find
\[
\cos (2 A)=
\]

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Answer

So, the final answer is $\cos (2 A) = \boxed{-0.6}$

Steps

Step 1 ：We are given that $\csc A=\frac{\sqrt{5}}{2}$, which implies $\sin A=\frac{2}{\sqrt{5}}$.

Step 2 ：Using the Pythagorean identity, we can find $\cos A$, which is $\sqrt{1-\sin^2 A}$. Since $A$ is in the first quadrant, $\cos A$ is positive.

Step 3 ：Then we can use the double angle formula for cosine, $\cos 2A = 1 - 2\sin^2 A$, to find $\cos 2A$.

Step 4 ：Substituting the values, we get $\cos 2A = -0.6$.

Step 5 ：So, the final answer is $\cos (2 A) = \boxed{-0.6}$