You can definitely use the identity \(1 + \cot^2(\theta) = \csc^2(\theta)\) to solve for \(\cot(\theta)\). Let's go through the steps.

### Step 1: Recall the identity.
The identity relating \(\cot(\theta)\) and \(\sin(\theta)\) is:

\[
1 + \cot^2(\theta) = \csc^2(\theta)
\]

Since \(\csc(\theta) = \frac{1}{\sin(\theta)}\), the identity can be rewritten as:

\[
1 + \cot^2(\theta) = \frac{1}{\sin^2(\theta)}
\]

### Step 2: Substitute the value of \(\sin(\theta)\).
Given \(\sin(\theta) = -\frac{\sqrt{7}}{4}\), we can substitute this into the equation:

\[
1 + \cot^2(\theta) = \frac{1}{\left(-\frac{\sqrt{7}}{4}\right)^2}
\]

### Step 3: Simplify the equation.
First, calculate \(\sin^2(\theta)\):

\[
\sin^2(\theta) = \left(-\frac{\sqrt{7}}{4}\right)^2 = \frac{7}{16}
\]

So,

\[
1 + \cot^2(\theta) = \frac{1}{\frac{7}{16}} = \frac{16}{7}
\]

### Step 4: Solve for \(\cot^2(\theta)\).
Now, subtract 1 from both sides:

\[
\cot^2(\theta) = \frac{16}{7} - 1 = \frac{16}{7} - \frac{7}{7} = \frac{9}{7}
\]

### Step 5: Take the square root.
Taking the square root of both sides gives:

\[
\cot(\theta) = \pm \sqrt{\frac{9}{7}} = \pm \frac{3}{\sqrt{7}}
\]

### Final Answer:
- If \(\theta\) is in the third quadrant, \(\cot(\theta) = \frac{3}{\sqrt{7}}\).
- If \(\theta\) is in the fourth quadrant, \(\cot(\theta) = -\frac{3}{\sqrt{7}}\).

This result matches the one we obtained earlier using \(\cos(\theta)\) and \(\sin(\theta)\).

Would you like to explore more details or ask further questions?

Here are 5 related questions you might find interesting:

1. How do you prove the identity \(1 + \cot^2(\theta) = \csc^2(\theta)\)?
2. What are some common trigonometric identities that involve \(\cot(\theta)\)?
3. How can you convert \(\cot(\theta)\) to \(\tan(\theta)\) and vice versa?
4. What is the significance of \(\csc(\theta)\) in trigonometry?
5. How do trigonometric identities help in simplifying complex trigonometric expressions?

**Tip:** When solving trigonometric equations, always consider both the principal values and the signs of the trigonometric functions based on the quadrant where the angle lies.

Explore how to use the identity 1 + cot^2(theta) = 1/sin^2(theta) in solving trigonometric equations. This detailed guide covers step-by-step solutions, including the application of trigonometric identities and formulas. Suitable for high school students studying trigonometry.

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