1. **State the problem:** Simplify the expression $$\sin\left(\frac{\pi}{2} - x\right) + \cos x \tan^2 x.$$ 2. **Recall the identity:** $$\sin\left(\frac{\pi}{2} - x\right) = \cos x.$$ This is a co-function identity in trigonometry. 3. **Rewrite the expression using the identity:** $$\cos x + \cos x \tan^2 x.$$ 4. **Factor out $$\cos x$$:** $$\cos x (1 + \tan^2 x).$$ 5. **Use the Pythagorean identity:** $$1 + \tan^2 x = \sec^2 x.$$ 6. **Substitute the identity:** $$\cos x \sec^2 x.$$ 7. **Recall that $$\sec x = \frac{1}{\cos x}$$, so:** $$\cos x \left(\frac{1}{\cos x}\right)^2 = \cos x \frac{1}{\cos^2 x}.$$ 8. **Simplify by canceling one $$\cos x$$:** $$\cancel{\cos x} \frac{1}{\cancel{\cos x} \cos x} = \frac{1}{\cos x} = \sec x.$$ **Final answer:** $$\sin\left(\frac{\pi}{2} - x\right) + \cos x \tan^2 x = \sec x.$$