Finding Inverse Trig Functions

I know this may sound elementary...but trig has always been a frustration of mine. What are some good tricks to know or techniques on figuring these out? I mean I know all the common angles in the first quadrant, but I'm not well versed after that. I think therein lies a problem. Is it just a matter of memorizing the unit circle, restricted domains of the inverses, and the following identities?

$\displaystyle

\begin{gathered}

\cos ^{ - 1} x = \pi /2 - \sin ^{ - 1} x \hfill \\

\cot ^{ - 1} x = \pi /2 - \tan ^{ - 1} x \hfill \\

\csc ^{ - 1} x = \pi /2 - \sec ^{ - 1} x \hfill \\

\sin ^{ - 1} ( - x) = - \sin ^{ - 1} (x) \hfill \\

\cos ^{ - 1} ( - x) = \pi - \cos ^{ - 1} x \hfill \\

\end{gathered}

$

For example. I know this is wrong but this what I tend to do.

$\displaystyle

\begin{gathered}

\cot ^{ - 1} ( - 1) \hfill \\

\cot \theta = - 1 \hfill \\

\frac{1}

{{\tan \theta }} = - 1 \hfill \\

\tan \theta = - 1 \hfill \\

\theta = - \pi /4 \hfill \\

\end{gathered}

$

Until I plug into the above formula, do I discover I'm wrong.

$\displaystyle

\begin{gathered}

\cot ^{ - 1} ( - 1) = \pi /2 + \tan ^{ - 1} 1 \hfill \\

\cot ^{ - 1} ( - 1) = \pi /2 + \pi /4 \hfill \\

\cot ^{ - 1} ( - 1) = 3\pi /4 \hfill \\

\end{gathered}

$

Is there any way to avoid these fallacies? or is it to just memorize the above?