Answer
$$\cos\theta=\frac{24}{25}$$
$$\tan\theta=-\frac{7}{24}$$
$$\cot\theta=-\frac{24}{7}$$
$$\sec\theta=\frac{25}{24}$$
$$\csc\theta=-\frac{25}{7}$$
Work Step by Step
$$\sin\theta=-\frac{7}{25}\hspace{1.5cm}\theta\hspace{.1cm}\text{in quadrant IV}$$
$\theta$ is in quadrant IV, meaning that $\cos\theta\gt0$.
1) First, apply Pythagorean Identity to find $\cos\theta$
$$\cos^2\theta=1-\sin^2\theta=1-\Big(-\frac{7}{25}\Big)^2$$
$$\cos^2\theta=1-\Big(\frac{49}{625}\Big)=\frac{576}{625}$$
So, $$\cos\theta=\pm\frac{24}{25}$$
But since $\cos\theta\gt0$, $$\cos\theta=\frac{24}{25}$$
2) Apply Quotient Identities to find $\tan\theta$ and $\cot\theta$
$$\tan\theta=\frac{\sin\theta}{\cos\theta}=\frac{-\frac{7}{25}}{\frac{24}{25}}=-\frac{7}{24}$$
$$\cot\theta=\frac{\cos\theta}{\sin\theta}=\frac{\frac{24}{25}}{-\frac{7}{25}}=-\frac{24}{7}$$
3) Apply Reciprocal Identities to find $\sec\theta$ and $\csc\theta$
$$\sec\theta=\frac{1}{\cos\theta}=\frac{1}{\frac{24}{25}}=\frac{25}{24}$$
$$\csc\theta=\frac{1}{\sin\theta}=\frac{1}{-\frac{7}{25}}=-\frac{25}{7}$$
