Solve the trigonometric identity:

Let's prove the given trigonometric identity: \[\frac{1}{1 + \cot^2 \alpha} + \cos^2 \alpha = 1\] We know the trigonometric identity: \[1 + \cot^2 \alpha = \csc^2 \alpha\] So, we can rewrite the expression as: \[\frac{1}{\csc^2 \alpha} + \cos^2 \alpha = 1\] We also know that \(\csc \alpha = \frac{1}{\sin \alpha}\), therefore, \(\csc^2 \alpha = \frac{1}{\sin^2 \alpha}\). Thus, we can write: \[\frac{1}{\frac{1}{\sin^2 \alpha}} + \cos^2 \alpha = 1\] Which simplifies to: \[\sin^2 \alpha + \cos^2 \alpha = 1\] This is a fundamental trigonometric identity, which is true. Thus, the original identity is proven. Therefore, the given trigonometric identity \(\frac{1}{1 + \cot^2 \alpha} + \cos^2 \alpha = 1\) is correct.