To determine the value of \(\sin\ t,\ \cos\ t\ \text{and}\ \tan\ t.\)

Given:

\(\displaystyle \csc{{\left({t}\right)}}={\left[\frac{{-{12}}}{{{7}}}\right]}{\quad\text{and}\quad}{\left[{\left(-\frac{\pi}{{2}}\right)}<{t}<{\left(\frac{\pi}{{2}}\right)}\right]}\)

\(\displaystyle \csc{{\left({t}\right)}}={\left[\frac{{-{12}}}{{{7}}}\right]}\)

Using trigonometric identity, \(\displaystyle \sin{{t}}=\frac{1}{ \csc{{t}}}\)

\(\displaystyle \sin{{t}}=\frac{1}{{\frac{{-{12}}}{{{7}}}}}\)

\(\displaystyle \sin{{t}}=-\frac{7}{{12}}\)

\(\displaystyle{{\cos}^{2}{t}}={1}-{{\sin}^{2}{t}}\)

\(\displaystyle \cos{{t}}=\pm\sqrt{{{1}-{\left(-\frac{7}{{12}}\right)}^{2}}}\)

\(\displaystyle \cos{{t}}=\pm\sqrt{{{1}-\frac{49}{{144}}}}\)

\(\displaystyle \cos{{t}}=\pm\sqrt{{\frac{95}{{144}}}}\)

Since, \(\displaystyle-\frac{\pi}{{2}}\le{t}\le\frac{\pi}{{2}}\)

\(\displaystyle\Rightarrow \cos{{t}}=\frac{\sqrt{{95}}}{{12}}\)

\(\displaystyle \tan{{t}}=\frac{{ \sin{{t}}}}{{ \cos{{t}}}}\)

\(\displaystyle \tan{{t}}=\frac{{{\left(-\frac{7}{{12}}\right)}}}{{{\left(\frac{\sqrt{{95}}}{{12}}\right)}}}\)

\(\displaystyle \tan{{t}}=-\frac{7}{\sqrt{{95}}}\)

Thus,

\(\displaystyle \sin{{t}}=-\frac{7}{{12}},\)

\(\displaystyle \cos{{t}}=\frac{\sqrt{{95}}}{{12}},\)

\(\displaystyle \tan{{t}}=-\frac{7}{\sqrt{{95}}}\)