Simplify: \( \tan x \cot x-\cos ^{2} x \)

To simplify the expression \( \tan x \cot x - \cos^2 x \), we can start by recalling the definitions of tangent and cotangent: \[ \tan x = \frac{\sin x}{\cos x} \quad \text{and} \quad \cot x = \frac{\cos x}{\sin x} \] Now, we can simplify the first part of the expression: \[ \tan x \cot x = \frac{\sin x}{\cos x} \cdot \frac{\cos x}{\sin x} = 1 \] So we can substitute this back into the original expression: \[ \tan x \cot x - \cos^2 x = 1 - \cos^2 x \] Next, we use the Pythagorean identity, which tells us that: \[ \sin^2 x + \cos^2 x = 1 \quad \Rightarrow \quad \sin^2 x = 1 - \cos^2 x \] Therefore, we replace \(1 - \cos^2 x\) with \(\sin^2 x\): \[ 1 - \cos^2 x = \sin^2 x \] Thus, the simplified form of the given expression is: \[ \sin^2 x \]