To prove the identity of the **tangent** function (\(\tan \theta\)), let's first define it using basic trigonometric terms. The tangent of an angle \(\theta\) in a right triangle is the ratio of the opposite side to the adjacent side.

We begin with the definition of the tangent function using **sine** and **cosine** functions:

\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\]

### Step-by-Step Proof:

#### 1. Start with a right triangle.
Consider a right triangle with an angle \(\theta\). Let the length of the side opposite to \(\theta\) be \(O\), the side adjacent to \(\theta\) be \(A\), and the hypotenuse be \(H\).

By definition, we have:

\[
\sin \theta = \frac{O}{H} \quad \text{(opposite over hypotenuse)}
\]
\[
\cos \theta = \frac{A}{H} \quad \text{(adjacent over hypotenuse)}
\]

#### 2. Derive the tangent.
Now, we know that the tangent function is the ratio of the opposite side to the adjacent side:

\[
\tan \theta = \frac{O}{A}
\]

#### 3. Express tangent in terms of sine and cosine.
Since:

\[
\sin \theta = \frac{O}{H} \quad \text{and} \quad \cos \theta = \frac{A}{H}
\]

we can substitute these into the equation for tangent:

\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\]

Thus, we've proved that:

\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\]

### Summary:
The proof follows directly from the geometric definition of sine and cosine in a right triangle and their relationship to the tangent function.

Would you like more details on this or have any questions?

Here are five questions to expand your understanding:
1. How does the tangent function behave as \(\theta \to 90^\circ\)?
2. What is the derivative of the tangent function?
3. How can we use the tangent function in solving triangles?
4. What are the values of \(\tan \theta\) for common angles like 30°, 45°, and 60°?
5. How does the unit circle relate to the tangent function?

**Tip:** Memorizing the relationships between sine, cosine, and tangent helps in understanding their interdependence, making trigonometric identities easier to manipulate.

Discover how to prove the trigonometric identity tan(θ) = sin(θ) / cos(θ) using right triangle geometry and the relationship between sine and cosine. Suitable for students in Grades 9-12, this step-by-step guide explains core trigonometric concepts and provides insights into fundamental identities.

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