Complete the identity. \( \tan x(\cot x-\cos x)=? \)

To complete the identity, let's simplify the expression step-by-step. Starting with the expression: \[ \tan x(\cot x - \cos x) \] We know: \[ \tan x = \frac{\sin x}{\cos x} \quad \text{and} \quad \cot x = \frac{\cos x}{\sin x} \] Substituting these definitions in, we have: \[ \tan x(\cot x - \cos x) = \frac{\sin x}{\cos x} \left(\frac{\cos x}{\sin x} - \cos x\right) \] Now simplify the term inside the parentheses: \[ \frac{\cos x}{\sin x} - \cos x = \frac{\cos x - \cos x \sin x}{\sin x} = \frac{\cos x(1 - \sin x)}{\sin x} \] Now substitute this back into the original expression: \[ \tan x \cdot \frac{\cos x(1 - \sin x)}{\sin x} = \frac{\sin x}{\cos x} \cdot \frac{\cos x(1 - \sin x)}{\sin x} \] The \(\sin x\) in the numerator and denominator cancels out, leaving us with: \[ \frac{\cancel{\sin x}}{\cos x} \cdot \frac{\cos x(1 - \sin x)}{\cancel{\sin x}} = 1 - \sin x \] Thus, the completed identity is: \[ \tan x(\cot x - \cos x) = 1 - \sin x \]