We are tasked to solve the trigonometric equation:

$\csc^2(\theta) = 7\cot(\theta) + 9$

Step 1: Recall Trigonometric Identities

The cosecant and cotangent functions have the identity: $\csc^2(\theta) = 1 + \cot^2(\theta)$

Substituting this into the equation, we get: $1 + \cot^2(\theta) = 7\cot(\theta) + 9$

Step 2: Rearrange the Equation

Rearrange the terms into a standard quadratic form: $\cot^2(\theta) - 7\cot(\theta) - 8 = 0$

Step 3: Solve the Quadratic Equation

Let $x = \cot(\theta)$, so the equation becomes: $x^2 - 7x - 8 = 0$

Solve using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 1$, $b = -7$, and $c = -8$. Substituting these values: $x = \frac{-(-7) \pm \sqrt{(-7)^2 - 4(1)(-8)}}{2(1)}$ $x = \frac{7 \pm \sqrt{49 + 32}}{2}$ $x = \frac{7 \pm \sqrt{81}}{2}$ $x = \frac{7 \pm 9}{2}$

Thus: $x = \frac{7 + 9}{2} = 8 \quad \text{or} \quad x = \frac{7 - 9}{2} = -1$

Step 4: Interpret Solutions

Recall that $x = \cot(\theta)$, so: $\cot(\theta) = 8 \quad \text{or} \quad \cot(\theta) = -1$

Case 1: $\cot(\theta) = 8$

This implies: $\theta = \cot^{-1}(8)$ Using a calculator: $\theta \approx 0.125 \, \text{radians (principal value)}.$

Since the cotangent function has a period of $\pi$, general solutions are: $\theta = 0.125 + k\pi, \quad k \in \mathbb{Z}.$

Case 2: $\cot(\theta) = -1$

This implies: $\theta = \cot^{-1}(-1)$ Using a calculator: $\theta = \frac{3\pi}{4} \, \text{radians (principal value)}.$

General solutions are: $\theta = \frac{3\pi}{4} + k\pi, \quad k \in \mathbb{Z}.$

Final Answer:

$\theta = 0.125 + k\pi, \, \frac{3\pi}{4} + k\pi, \quad k \in \mathbb{Z}.$

Let me know if you would like further clarification or detailed explanations!

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5 Relative Questions:

1. How do we handle periodicity in trigonometric functions?
2. What is the general solution for $\cot(\theta) = x$?
3. How do trigonometric identities simplify complex equations?
4. Why does the quadratic formula help in solving trigonometric equations?
5. How does the unit circle relate to cotangent values?

Tip: Always check the domain restrictions for trigonometric functions (e.g., $\csc(\theta)$ is undefined for $\theta = n\pi$).