If `sectheta=-13//5,` where `90^(@) lt theta lt 180^(@),` evalute : `sin2theta,cos2thetaand tan2theta.`

Answer

To solve the problem, we need to evaluate \( \sin 2\theta \), \( \cos 2\theta \), and \( \tan 2\theta \) given that \( \sec \theta = -\frac{13}{5} \) and \( 90^\circ < \theta < 180^\circ \).

### Step 1: Understand the Quadrant
Since \( \theta \) is in the second quadrant, we know that:
- \( \sin \theta \) is positive.
- \( \cos \theta \) is negative.
- \( \sec \theta \) is negative (which it is, as given).

### Step 2: Find \( \cos \theta \)
Given \( \sec \theta = -\frac{13}{5} \), we can find \( \cos \theta \):
\[ \cos \theta = \frac{1}{\sec \theta} = \frac{1}{-\frac{13}{5}} = -\frac{5}{13} \]

### Step 3: Use Pythagorean Identity to Find \( \sin \theta \)
Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ \sin^2 \theta + \left(-\frac{5}{13}\right)^2 = 1 \]
\[ \sin^2 \theta + \frac{25}{169} = 1 \]
\[ \sin^2 \theta = 1 - \frac{25}{169} = \frac{169 - 25}{169} = \frac{144}{169} \]
\[ \sin \theta = \sqrt{\frac{144}{169}} = \frac{12}{13} \]
Since \( \theta \) is in the second quadrant, \( \sin \theta \) is positive:
\[ \sin \theta = \frac{12}{13} \]

### Step 4: Calculate \( \sin 2\theta \)
Using the double angle formula:
\[ \sin 2\theta = 2 \sin \theta \cos \theta \]
Substituting the values:
\[ \sin 2\theta = 2 \cdot \frac{12}{13} \cdot \left(-\frac{5}{13}\right) = -\frac{120}{169} \]

### Step 5: Calculate \( \cos 2\theta \)
Using the double angle formula:
\[ \cos 2\theta = 2 \cos^2 \theta - 1 \]
Substituting the value of \( \cos \theta \):
\[ \cos 2\theta = 2 \left(-\frac{5}{13}\right)^2 - 1 = 2 \cdot \frac{25}{169} - 1 = \frac{50}{169} - 1 = \frac{50 - 169}{169} = -\frac{119}{169} \]

### Step 6: Calculate \( \tan 2\theta \)
Using the formula:
\[ \tan 2\theta = \frac{\sin 2\theta}{\cos 2\theta} \]
Substituting the values:
\[ \tan 2\theta = \frac{-\frac{120}{169}}{-\frac{119}{169}} = \frac{120}{119} \]

### Final Answers
Thus, we have:
\[ \sin 2\theta = -\frac{120}{169}, \quad \cos 2\theta = -\frac{119}{169}, \quad \tan 2\theta = \frac{120}{119} \]