1. **State the problem:** Simplify and verify the identity: $$\frac{\cos^2 a + \cot a}{\cos^2 a - \cot a} = \frac{\cos^2 a \tan a + 1}{\cos^2 a \tan a - 1}$$ 2. **Recall important formulas and identities:** - $\cot a = \frac{\cos a}{\sin a}$ - $\tan a = \frac{\sin a}{\cos a}$ - Pythagorean identity: $\sin^2 a + \cos^2 a = 1$ 3. **Simplify the left-hand side (LHS):** $$\frac{\cos^2 a + \cot a}{\cos^2 a - \cot a} = \frac{\cos^2 a + \frac{\cos a}{\sin a}}{\cos^2 a - \frac{\cos a}{\sin a}}$$ Multiply numerator and denominator by $\sin a$ to clear denominators: $$= \frac{\cos^2 a \sin a + \cos a}{\cos^2 a \sin a - \cos a}$$ 4. **Simplify the right-hand side (RHS):** $$\frac{\cos^2 a \tan a + 1}{\cos^2 a \tan a - 1} = \frac{\cos^2 a \cdot \frac{\sin a}{\cos a} + 1}{\cos^2 a \cdot \frac{\sin a}{\cos a} - 1} = \frac{\cos a \sin a + 1}{\cos a \sin a - 1}$$ 5. **Compare LHS and RHS:** LHS numerator: $\cos^2 a \sin a + \cos a = \cos a (\cos a \sin a + 1)$ LHS denominator: $\cos^2 a \sin a - \cos a = \cos a (\cos a \sin a - 1)$ So LHS: $$\frac{\cos a (\cos a \sin a + 1)}{\cos a (\cos a \sin a - 1)} = \frac{\cos a \sin a + 1}{\cos a \sin a - 1}$$ Since $\cos a \neq 0$, we can cancel $\cos a$. This matches the RHS exactly. 6. **Conclusion:** The identity holds true. **Final answer:** $$\frac{\cos^2 a + \cot a}{\cos^2 a - \cot a} = \frac{\cos^2 a \tan a + 1}{\cos^2 a \tan a - 1}$$ is verified.