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The problem presents an equation involving trigonometric identities and asks to find the value of the angle \( x \) for a ramp with a given equation. The equation is:

\[
\sec^2 x \cdot \csc^2 x - \sec^2 x = 4
\]

### Step 1: Simplify the given equation

We start by breaking down the given equation:

\[
\sec^2 x \cdot \csc^2 x - \sec^2 x = 4
\]

The secant and cosecant identities are related to sine and cosine:
\[
\sec x = \frac{1}{\cos x} \quad \text{and} \quad \csc x = \frac{1}{\sin x}
\]
Thus,
\[
\sec^2 x = \frac{1}{\cos^2 x} \quad \text{and} \quad \csc^2 x = \frac{1}{\sin^2 x}
\]

Substituting these into the original equation:

\[
\left(\frac{1}{\cos^2 x} \cdot \frac{1}{\sin^2 x}\right) - \frac{1}{\cos^2 x} = 4
\]

### Step 2: Simplify the product
\[
\frac{1}{\cos^2 x \sin^2 x} - \frac{1}{\cos^2 x} = 4
\]

### Step 3: Combine the terms

Factor out \( \frac{1}{\cos^2 x} \):

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

### Step 4: Use the identity for \( \sin^2 x \)

We know that \( 1 - \sin^2 x = \cos^2 x \), so:

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

This simplifies to:

\[
\frac{1}{\sin^2 x} = 4
\]

### Step 5: Solve for \( \sin^2 x \)

\[
\sin^2 x = \frac{1}{4}
\]

Taking the square root of both sides:

\[
\sin x = \frac{1}{2}
\]

### Step 6: Determine the angle \( x \)

The angle whose sine is \( \frac{1}{2} \) is \( x = 30^\circ \).

Thus, the value of \( x \) is:

\[
\boxed{30^\circ}
\]

Let me know if you would like further explanation on any of the steps!

---

Here are 5 related questions:
1. How can we derive the secant and cosecant identities from basic trigonometric functions?
2. What is the relationship between sine, cosine, and their respective reciprocal identities (secant and cosecant)?
3. Why is the identity \( 1 - \sin^2 x = \cos^2 x \) important in trigonometric simplifications?
4. How do you solve equations that involve multiple trigonometric identities?
5. How can you apply these trigonometric concepts in real-world problems like the construction of ramps?

**Tip:** Whenever you encounter a trigonometric identity equation, always try to simplify using basic identities like \( \sin^2 x + \cos^2 x = 1 \). This often leads to a more manageable equation.

Se desea construir la rampa mostrada, para lo cual es indispensable saber cuánto mide el ángulo x. Halle el valor de dicho ángulo si se cumple que: Sec²x · Csc²x − Sec²x = 4

Solve the trigonometric equation sec²x · csc²x − sec²x = 4 to find the angle x for a ramp. This problem involves key trigonometric concepts and simplifications using secant and cosecant identities. Suitable for Grades 10-12.

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