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 trigonometric identities. First, we know that: \[ \cot\left(\frac{\pi}{2}-x\right) = \tan x \] Thus, \[ \cot^{2}\left(\frac{\pi}{2}-x\right) = \tan^{2} x \] Next, we also have: \[ \cos\left(\frac{\pi}{2}-x\right) = \sin x \] Which means: \[ \cos^{2}\left(\frac{\pi}{2}-x\right) = \sin^{2} x \] Putting it all back together, our expression transforms into: \[ \tan^{2} x + \sin^{2} x + \cos^{2} x \] Utilizing the Pythagorean identity \(\sin^{2} x + \cos^{2} x = 1\), we substitute: \[ \tan^{2} x + 1 \] And since \(\tan^{2} x + 1 = \sec^{2} x\), we finally obtain: \[ \sec^{2} x \] Thus, the simplified expression is: \[ \sec^{2} x \]