\(4\tan ^{(2)}\theta =9\sec ^{(2)}\theta \)

We are given the trigonometric equation:

\[
4 \tan^2 \theta = 9 \sec^2 \theta
\]

### Step 1: Express everything in terms of \(\sin \theta\) and \(\cos \theta\)

Recall the following trigonometric identities:
- \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
- \(\sec \theta = \frac{1}{\cos \theta}\)

Thus:
\[
\tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta}
\quad \text{and} \quad 
\sec^2 \theta = \frac{1}{\cos^2 \theta}
\]

Substitute these into the original equation:

\[
4 \left( \frac{\sin^2 \theta}{\cos^2 \theta} \right) = 9 \left( \frac{1}{\cos^2 \theta} \right)
\]

### Step 2: Simplify the equation

Both sides of the equation have \(\frac{1}{\cos^2 \theta}\), so we can cancel them out:

\[
4 \sin^2 \theta = 9
\]

### Step 3: Solve for \(\sin^2 \theta\)

\[
\sin^2 \theta = \frac{9}{4}
\]

### Step 4: Solve for \(\sin \theta\)

Take the square root of both sides:

\[
\sin \theta = \pm \frac{3}{2}
\]

However, since \(\sin \theta\) must lie between \(-1\) and \(1\), this equation has **no real solutions**. Therefore, there are no real solutions for \(\theta\) that satisfy the given equation.

---

Would you like more details, or do you have any other questions?

Here are 5 questions related to this:

1. What is the domain of the sine function?
2. How does the range of \(\sin \theta\) limit solutions in trigonometric equations?
3. How would this problem change if we replaced \(\sin^2 \theta\) with \(\cos^2 \theta\)?
4. What are the possible solutions if this were an equation involving complex numbers?
5. How do the graphs of \(\tan^2 \theta\) and \(\sec^2 \theta\) compare?

**Tip:** Always check the range of trigonometric functions to determine if a solution is possible.

Explore the step-by-step solution of the trigonometric equation 4tan^2θ = 9sec^2θ using core trigonometric identities. Learn how to simplify and solve the equation to determine if real solutions exist. Suitable for students in Grades 10-12.

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