https://file.aiquickdraw.com/mathbot/file/17263277845240bm6631p.jpg

The equation in the image is:

\[
\sec^2 \theta + 3 \csc^2 \theta = 8
\]

Let’s solve for \(\theta\).

### Step 1: Recall the basic trigonometric identities.
- \(\sec^2 \theta = 1 + \tan^2 \theta\)
- \(\csc^2 \theta = 1 + \cot^2 \theta\)

These identities can be substituted into the equation to simplify.

### Step 2: Substituting the identities
\[
(1 + \tan^2 \theta) + 3(1 + \cot^2 \theta) = 8
\]

### Step 3: Expand the equation
\[
1 + \tan^2 \theta + 3 + 3 \cot^2 \theta = 8
\]
\[
\tan^2 \theta + 3 \cot^2 \theta + 4 = 8
\]

### Step 4: Simplify the equation
\[
\tan^2 \theta + 3 \cot^2 \theta = 4
\]

### Step 5: Express \(\cot^2 \theta\) in terms of \(\tan^2 \theta\)
We know that:
\[
\cot^2 \theta = \frac{1}{\tan^2 \theta}
\]

So, the equation becomes:
\[
\tan^2 \theta + \frac{3}{\tan^2 \theta} = 4
\]

### Step 6: Multiply the equation by \(\tan^2 \theta\) to eliminate the denominator
Let \(x = \tan^2 \theta\). Then, we get:
\[
x^2 + 3 = 4x
\]
\[
x^2 - 4x + 3 = 0
\]

### Step 7: Solve the quadratic equation
\[
(x - 3)(x - 1) = 0
\]
Thus, \(x = 3\) or \(x = 1\).

### Step 8: Solve for \(\theta\)
Since \(x = \tan^2 \theta\), we have two cases:
- If \(\tan^2 \theta = 3\), then \(\tan \theta = \sqrt{3}\), which gives \(\theta = 60^\circ\) or \(\theta = 240^\circ\).
- If \(\tan^2 \theta = 1\), then \(\tan \theta = 1\), which gives \(\theta = 45^\circ\) or \(\theta = 225^\circ\).

Thus, the possible values of \(\theta\) are \(45^\circ\), \(60^\circ\), \(225^\circ\), or \(240^\circ\).

---

Would you like more details or clarifications?

Here are 5 related questions:
1. What is the general solution for \(\tan \theta = 1\)?
2. How would the equation change if the constant on the right-hand side were different?
3. Can you solve this equation graphically using trigonometric curves?
4. What other identities might help simplify equations involving \(\sec^2 \theta\) and \(\csc^2 \theta\)?
5. How can you verify the solutions of a trigonometric equation using a calculator?

**Tip**: Always remember to check if your solutions lie within the required domain when solving trigonometric equations.

Solve the equation: sec^2(θ) + 3csc^2(θ) = 8

This detailed solution shows how to solve sec^2(θ) + 3csc^2(θ) = 8 using fundamental trigonometric identities and algebra. This problem demonstrates the use of Pythagorean identities and quadratic equations. Suitable for high school students (Grades 10-12).

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