A function value and a quadrant are given. Find the other five function values.

A function value and a quadrant are given. Find the other five function values. Give exact answers, uing radicals as needed. Rationalize all denominators.
$$\displaystyle{\cot{\theta}}=-{4}$$, Quandrant 4

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Arnold Odonnell
It is given that, $$\displaystyle{\cot{\theta}}=-{4}$$
Compute the other five functional values as follows.
$$\displaystyle{{\csc}^{{2}}\theta}-{{\cot}^{{2}}\theta}={1}$$
$$\displaystyle{\csc{\theta}}=\sqrt{{{1}+{{\cot}^{{2}}\theta}}}$$
$$\displaystyle{\csc{\theta}}=\sqrt{{{1}+{\left(-{4}\right)}^{{2}}}}$$
$$\displaystyle{\csc{\theta}}=-\sqrt{{{17}}}$$
Compute the value of $$\displaystyle{\sin{\theta}}$$ as follows.
$$\displaystyle{\sin{\theta}}={\frac{{{1}}}{{{\csc{\theta}}}}}$$
$$\displaystyle{\sin{\theta}}=-{\frac{{{1}}}{{\sqrt{{{17}}}}}}$$
$$\displaystyle{\sin{\theta}}=-{\frac{{\sqrt{{{17}}}}}{{{17}}}}$$
Compute the value of $$\displaystyle{\cos{\theta}}$$ as follows.
$$\displaystyle{\cot{\theta}}={\frac{{{\cos{\theta}}}}{{{\sin{\theta}}}}}$$
$$\displaystyle{\cos{\theta}}={\cot{\theta}}\cdot{\sin{\theta}}$$
$$\displaystyle{\cos{\theta}}=-{4}{\left(-{\frac{{\sqrt{{{17}}}}}{{{17}}}}\right)}$$
$$\displaystyle{\cos{\theta}}={\frac{{{4}\sqrt{{{17}}}}}{{{17}}}}$$
Compute the value of $$\displaystyle{\sec{\theta}}$$ as follows.
$$\displaystyle{\sec{\theta}}={\frac{{{1}}}{{{\cos{\theta}}}}}$$
$$\displaystyle{\sec{\theta}}={\frac{{{17}}}{{{4}\sqrt{{{17}}}}}}$$
$$\displaystyle{\sec{\theta}}={\frac{{\sqrt{{{17}}}}}{{{4}}}}$$
Compute the value of $$\displaystyle{\tan{\theta}}$$ as follows.
$$\displaystyle{\tan{\theta}}={\frac{{{1}}}{{{\cot{\theta}}}}}$$
$$\displaystyle{\tan{\theta}}=-{\frac{{{1}}}{{{4}}}}$$