1. **State the problem:** Prove the identity $$\frac{\cos x}{\sin x} + \frac{\sin x}{\cos x} = 2 \sin 2x$$. 2. **Recall formulas and rules:** - The double angle formula for sine: $$\sin 2x = 2 \sin x \cos x$$. - To add fractions, find a common denominator. 3. **Start with the left-hand side (LHS):** $$\frac{\cos x}{\sin x} + \frac{\sin x}{\cos x}$$ 4. **Find common denominator and combine:** $$\frac{\cos x \cdot \cos x}{\sin x \cdot \cos x} + \frac{\sin x \cdot \sin x}{\cos x \cdot \sin x} = \frac{\cos^2 x + \sin^2 x}{\sin x \cos x}$$ 5. **Use Pythagorean identity:** $$\cos^2 x + \sin^2 x = 1$$ So, $$\frac{1}{\sin x \cos x}$$ 6. **Rewrite the right-hand side (RHS):** $$2 \sin 2x = 2 \cdot 2 \sin x \cos x = 4 \sin x \cos x$$ 7. **Check equality:** LHS is $$\frac{1}{\sin x \cos x}$$ but RHS is $$4 \sin x \cos x$$, so they are not equal. 8. **Re-examine the problem statement:** The original identity is likely misstated. Possibly it should be: $$\frac{\cos x}{\sin x} + \frac{\sin x}{\cos x} = 2 \csc 2x$$ or $$\frac{\cos x}{\sin x} + \frac{\sin x}{\cos x} = 2 \sec 2x$$. 9. **Alternatively, simplify the LHS:** $$\frac{\cos x}{\sin x} + \frac{\sin x}{\cos x} = \cot x + \tan x$$ Using the identity: $$\cot x + \tan x = \frac{2}{\sin 2x}$$ 10. **Therefore, the correct identity is:** $$\frac{\cos x}{\sin x} + \frac{\sin x}{\cos x} = \frac{2}{\sin 2x}$$ **Final answer:** The original identity as stated is incorrect. The correct identity is $$\frac{\cos x}{\sin x} + \frac{\sin x}{\cos x} = \frac{2}{\sin 2x}$$.