4.1 Without the use of tables or a calculator prove that $$ \frac{1-\cos 2 A}{\sin 2 A}= an $$

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4.1 Without the use of tables or a calculator prove that $$ \frac{1-\cos 2 A}{\sin 2 A}=	an $$

UpStudy AI · Accepted Answer

To prove the trigonometric identity:

$$
\frac{1 - \cos 2A}{\sin 2A} = \tan A
$$

**without using tables or a calculator**, we'll utilize fundamental trigonometric identities. Here's a step-by-step demonstration:

### **Step 1: Recall the Double-Angle Formulas**

The double-angle formulas for sine and cosine are essential for this proof:

1. **Sine Double-Angle Identity**:
   $$
   \sin 2A = 2 \sin A \cos A
   $$

2. **Cosine Double-Angle Identity**:
   $$
   \cos 2A = 1 - 2 \sin^2 A
   $$
   
   *(Alternatively, $$\cos 2A = 2 \cos^2 A - 1$$ can also be used, but we'll proceed with the first form.)*

### **Step 2: Substitute the Cosine Double-Angle Identity into the Numerator**

Starting with the numerator of the left-hand side (LHS):

$$
1 - \cos 2A = 1 - (1 - 2 \sin^2 A) = 2 \sin^2 A
$$

So, the expression becomes:

$$
\frac{1 - \cos 2A}{\sin 2A} = \frac{2 \sin^2 A}{\sin 2A}
$$

### **Step 3: Substitute the Sine Double-Angle Identity into the Denominator**

Replace $$\sin 2A$$ with its double-angle equivalent:

$$
\frac{2 \sin^2 A}{\sin 2A} = \frac{2 \sin^2 A}{2 \sin A \cos A}
$$

### **Step 4: Simplify the Fraction**

Cancel out the common factors in the numerator and the denominator:

$$
\frac{2 \sin^2 A}{2 \sin A \cos A} = \frac{\sin^2 A}{\sin A \cos A} = \frac{\sin A}{\cos A}
$$

*(Here, $$2$$ in the numerator and denominator cancels out, and one $$\sin A$$ in the numerator cancels with one $$\sin A$$ in the denominator.)*

### **Step 5: Recognize the Tangent Function**

The simplified expression is the definition of the tangent function:

$$
\frac{\sin A}{\cos A} = \tan A
$$

### **Conclusion**

Thus, we've shown that:

$$
\frac{1 - \cos 2A}{\sin 2A} = \tan A
$$

This completes the proof.
$$ \frac{1 - \cos 2A}{\sin 2A} = \tan A $$

4.1 Without the use of tables or a calculator prove that \( \frac{1-\cos 2 A}{\sin 2 A}=\tan \)

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