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

Question

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

UpStudy AI · Accepted Answer

Certainly! Let's prove the trigonometric identity:

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

**Step 1: Recall Double Angle Identities**

First, recall the double angle identities for sine and cosine:

$$
\cos 2A = 1 - 2\sin^2 A \quad \text{and} \quad \sin 2A = 2\sin A \cos A
$$

**Step 2: Substitute the Double Angle Identity for $$\cos 2A$$**

Substitute $$\cos 2A = 1 - 2\sin^2 A$$ into the left-hand side (LHS) of the equation:

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

**Step 3: Substitute the Double Angle Identity for $$\sin 2A$$**

Now, substitute $$\sin 2A = 2\sin A \cos A$$ into the equation:

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

**Step 4: Simplify the Expression**

Cancel out $$\sin A$$ from the numerator and denominator:

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

**Conclusion**

Thus, we have shown that:

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

**Final Identity:**

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

Prove \( \frac{1-\cos 2 A}{\sin 2 A}=\tan \)