cos 2x + 1 / sin 2x = cot x

Steps to Solve

1. Rewrite cotangent in terms of sine and cosine

Recall that the cotangent function can be expressed as: $$ \cot x = \frac{\cos x}{\sin x} $$

2. Cross-multiply

Rearranging the equation gives: $$ \cos 2x + 1 = \cot x \cdot \sin 2x $$

So we can rewrite it as: $$ \cos 2x + 1 = \frac{\cos x}{\sin x} \cdot \sin 2x $$

3. Substitute using the double angle identity for sine

Using the identity $\sin 2x = 2 \sin x \cos x$: $$ \cos 2x + 1 = \frac{\cos x}{\sin x} \cdot 2 \sin x \cos x $$

This simplifies to: $$ \cos 2x + 1 = 2 \cos^2 x $$

4. Use the double angle identity for cosine

Recall that $\cos 2x = 2 \cos^2 x - 1$. Substitute this into the equation: $$ 2 \cos^2 x - 1 + 1 = 2 \cos^2 x $$

This simplifies to: $$ 2 \cos^2 x = 2 \cos^2 x $$

5. Conclude

Since we have demonstrated an identity, it means the original equation holds for any $x$ where both sides are defined.