Given that:

• $\csc \theta = -\frac{5}{4}$
• $\tan \theta > 0$

We can find the values of $\cot \theta$ and $\sec \theta$.

Step 1: Understand the properties of trigonometric functions

• $\csc \theta = \frac{1}{\sin \theta}$

  This gives: $\sin \theta = \frac{1}{\csc \theta} = \frac{1}{-\frac{5}{4}} = -\frac{4}{5}$

• Since $\tan \theta > 0$, we know that both $\sin \theta$ and $\cos \theta$ must have the same sign (either both positive or both negative). Given that $\sin \theta$ is negative, $\cos \theta$ must also be negative.

Step 2: Use the Pythagorean identity

The identity for sine and cosine is: $\sin^2 \theta + \cos^2 \theta = 1$

Substitute $\sin \theta = -\frac{4}{5}$: $\left( -\frac{4}{5} \right)^2 + \cos^2 \theta = 1$ $\frac{16}{25} + \cos^2 \theta = 1$ $\cos^2 \theta = 1 - \frac{16}{25} = \frac{25}{25} - \frac{16}{25} = \frac{9}{25}$ \cos \theta = -\frac{3}{5} \quad \text{(since \cos \theta is negative in this case)}

Step 3: Find $\cot \theta$ and $\sec \theta$

• $\cot \theta = \frac{\cos \theta}{\sin \theta}$ $\cot \theta = \frac{-\frac{3}{5}}{-\frac{4}{5}} = \frac{3}{4}$

• $\sec \theta = \frac{1}{\cos \theta}$ $\sec \theta = \frac{1}{-\frac{3}{5}} = -\frac{5}{3}$

Final Answer:

• $\cot \theta = \frac{3}{4}$
• $\sec \theta = -\frac{5}{3}$

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Would you like more details or clarification? Here are some related questions:

1. How can you verify the values of trigonometric functions using the unit circle?
2. What are the other trigonometric identities involving $\cot$ and $\sec$?
3. How do you determine the quadrant of an angle given the signs of the trigonometric functions?
4. Can you derive the values of trigonometric functions using the values of $\sin \theta$ and $\cos \theta$?
5. What happens to these functions if $\tan \theta < 0$?

Tip: Always check the quadrant where the angle lies by looking at the signs of the trigonometric functions.