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To derive the double-angle identity for \( \tan(2x) \), start with the definition \( \tan(2x) = \frac{\sin(2x)}{\cos(2x)} \). Use the double-angle formulas \( \sin(2x) = 2\sin(x)\cos(x) \) and \( \cos(2x) = \cos^2(x) - \sin^2(x) \). Substitute these into the tangent expression and simplify by dividing numerator and denominator by \( \cos^2(x) \) to get \( \tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)} \).
Solución
Deriving the double-angle identity for \( \tan(2x) \) using the definitions of sine and cosine involves several systematic steps. Here's a comprehensive guide to the derivation:
### **1. Start with the Definition of the Tangent Function**
The tangent of an angle is defined as the ratio of the sine to the cosine of that angle:
\[
\tan(2x) = \frac{\sin(2x)}{\cos(2x)}
\]
### **2. Apply the Double-Angle Formulas for Sine and Cosine**
To express \( \sin(2x) \) and \( \cos(2x) \) in terms of \( \sin(x) \) and \( \cos(x) \), use the double-angle identities:
\[
\sin(2x) = 2\sin(x)\cos(x)
\]
\[
\cos(2x) = \cos^2(x) - \sin^2(x)
\]
*Alternatively, the cosine double-angle identity can also be expressed as:*
\[
\cos(2x) = 1 - 2\sin^2(x) \quad \text{or} \quad \cos(2x) = 2\cos^2(x) - 1
\]
However, for this derivation, we'll use \( \cos(2x) = \cos^2(x) - \sin^2(x) \).
### **3. Substitute the Double-Angle Identities Back into the Tangent Expression**
Replace \( \sin(2x) \) and \( \cos(2x) \) in the expression for \( \tan(2x) \):
\[
\tan(2x) = \frac{2\sin(x)\cos(x)}{\cos^2(x) - \sin^2(x)}
\]
### **4. Express the Result in Terms of \( \tan(x) \)**
To simplify the expression and express it solely in terms of \( \tan(x) \), divide both the numerator and the denominator by \( \cos^2(x) \):
\[
\tan(2x) = \frac{2\sin(x)\cos(x)/\cos^2(x)}{(\cos^2(x) - \sin^2(x))/\cos^2(x)}
\]
Simplify each part:
- **Numerator:**
\[
\frac{2\sin(x)\cos(x)}{\cos^2(x)} = 2\sin(x)\frac{\cos(x)}{\cos^2(x)} = 2\frac{\sin(x)}{\cos(x)} = 2\tan(x)
\]
- **Denominator:**
\[
\frac{\cos^2(x) - \sin^2(x)}{\cos^2(x)} = \frac{\cos^2(x)}{\cos^2(x)} - \frac{\sin^2(x)}{\cos^2(x)} = 1 - \tan^2(x)
\]
### **5. Combine the Simplified Numerator and Denominator**
Now, substitute the simplified numerator and denominator back into the expression:
\[
\tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)}
\]
### **Final Double-Angle Identity for Tangent**
Thus, the double-angle identity for tangent is:
\[
\tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)}
\]
### **Summary of Steps**
1. **Start with the definition:** \( \tan(2x) = \frac{\sin(2x)}{\cos(2x)} \).
2. **Apply double-angle formulas:** \( \sin(2x) = 2\sin(x)\cos(x) \) and \( \cos(2x) = \cos^2(x) - \sin^2(x) \).
3. **Substitute these into the tangent expression:** \( \tan(2x) = \frac{2\sin(x)\cos(x)}{\cos^2(x) - \sin^2(x)} \).
4. **Express in terms of \( \tan(x) \) by dividing numerator and denominator by \( \cos^2(x) \):** \( \tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)} \).
This derivation shows how the double-angle identity for tangent is directly obtained from the fundamental definitions and properties of sine and cosine functions.
Revisado y aprobado por el equipo de tutoría de UpStudy
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