1. **State the problem:** Simplify the expression $$\sin\left(\frac{\pi}{2} - x\right) + \cos x \tan^2 x.$$\n\n2. **Recall trigonometric identities:**\n- $$\sin\left(\frac{\pi}{2} - x\right) = \cos x$$ (co-function identity).\n- $$\tan x = \frac{\sin x}{\cos x}$$ so $$\tan^2 x = \frac{\sin^2 x}{\cos^2 x}.$$\n\n3. **Rewrite the expression using these identities:**\n$$\sin\left(\frac{\pi}{2} - x\right) + \cos x \tan^2 x = \cos x + \cos x \cdot \frac{\sin^2 x}{\cos^2 x}.$$\n\n4. **Factor and simplify:**\n$$= \cos x + \frac{\cos x \sin^2 x}{\cos^2 x} = \cos x + \frac{\cancel{\cos x} \sin^2 x}{\cancel{\cos^2 x} \cos x} = \cos x + \frac{\sin^2 x}{\cos x}.$$\n\n5. **Combine terms over common denominator:**\n$$= \frac{\cos^2 x}{\cos x} + \frac{\sin^2 x}{\cos x} = \frac{\cos^2 x + \sin^2 x}{\cos x}.$$\n\n6. **Use Pythagorean identity:**\n$$\cos^2 x + \sin^2 x = 1,$$ so\n$$= \frac{1}{\cos x} = \sec x.$$\n\n**Final answer:** $$\sec x.$$