What steps would you follow to derive the double-angle identity for $$ an(2x) $$ from the definitions of sine and cosine?

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What steps would you follow to derive the double-angle identity for $$ 	an(2x) $$ from the definitions of sine and cosine?

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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.
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 $$ \tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)} $$.

What steps would you follow to derive the double-angle identity for \( \tan(2x) \) from the definitions of sine and cosine?

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