2. \( \tan ^{2} \theta\left(\cot ^{2} \theta-\cos ^{2} \theta\right)= \)
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To simplify the expression \( \tan ^{2} \theta\left(\cot ^{2} \theta-\cos ^{2} \theta\right) \), we start by substituting the definitions of tangent and cotangent in terms of sine and cosine: 1. Recall that \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Therefore, we have: \[ \tan^2 \theta = \frac{\sin^2 \theta}{\cos^2 \theta} \] \[ \cot^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta} \] 2. Now, substitute these into the expression: \[ \tan^{2} \theta \left( \cot^{2} \theta - \cos^{2} \theta \right) = \frac{\sin^2 \theta}{\cos^2 \theta} \left( \frac{\cos^2 \theta}{\sin^2 \theta} - \cos^2 \theta \right) \] 3. Simplify the term inside the parentheses: \[ \frac{\cos^2 \theta}{\sin^2 \theta} - \cos^2 \theta = \frac{\cos^2 \theta - \cos^2 \theta \sin^2 \theta}{\sin^2 \theta} = \frac{\cos^2 \theta (1 - \sin^2 \theta)}{\sin^2 \theta} = \frac{\cos^2 \theta \cos^2 \theta}{\sin^2 \theta} = \frac{\cos^4 \theta}{\sin^2 \theta} \] 4. Substitute this back into the expression: \[ \frac{\sin^2 \theta}{\cos^2 \theta} \cdot \frac{\cos^4 \theta}{\sin^2 \theta} = \frac{\cos^4 \theta}{\cos^2 \theta} = \cos^2 \theta \] So, the final simplified result is: \[ \tan^{2} \theta\left(\cot^{2} \theta-\cos^{2} \theta\right) = \cos^2 \theta \]
