Simplify. \( \cot ^{2}\left(\frac{\pi}{2}-x\right)+\cos ^{2}\left(\frac{\pi}{2}-x\right)+\cos ^{2} x \) \( [?]^{2} x \)

To simplify the expression \( \cot ^{2}\left(\frac{\pi}{2}-x\right)+\cos ^{2}\left(\frac{\pi}{2}-x\right)+\cos ^{2} x \), we can use some trigonometric identities. First, recall that: - \( \cot\left(\frac{\pi}{2}-x\right) = \tan(x) \), so \( \cot^{2}\left(\frac{\pi}{2}-x\right) = \tan^{2}(x) \). - \( \cos\left(\frac{\pi}{2}-x\right) = \sin(x) \), so \( \cos^{2}\left(\frac{\pi}{2}-x\right) = \sin^{2}(x) \). Now, substituting these identities into the expression gives us: \[ \tan^{2}(x) + \sin^{2}(x) + \cos^{2}(x). \] Recall the Pythagorean identity \( \sin^{2}(x) + \cos^{2}(x) = 1 \). Thus, we can replace \( \sin^{2}(x) + \cos^{2}(x) \) with 1: \[ \tan^{2}(x) + 1. \] Using the identity for tangent, we also know that: \[ \tan^{2}(x) + 1 = \sec^{2}(x). \] Therefore, the entire expression simplifies to: \[ \sec^{2}(x). \] Thus, the final answer is: \[ \sec^{2}(x). \]