Prove and verify: a) \( \tan \varnothing \cot ^{2} \varnothing=\cot \varnothing \)

To prove \( \tan \varnothing \cot^{2} \varnothing = \cot \varnothing \), we start by recalling the definitions of the trigonometric functions involved. We know that \( \tan \varnothing = \frac{\sin \varnothing}{\cos \varnothing} \) and \( \cot \varnothing = \frac{\cos \varnothing}{\sin \varnothing} \). Therefore, we can substitute \( \cot \varnothing \) into the equation: 1. Rewrite \( \cot^{2} \varnothing \) as \( \left( \frac{\cos \varnothing}{\sin \varnothing} \right)^{2} = \frac{\cos^{2} \varnothing}{\sin^{2} \varnothing} \). Now substitute into the left-hand side of the equation: \[ \tan \varnothing \cot^{2} \varnothing = \tan \varnothing \cdot \frac{\cos^{2} \varnothing}{\sin^{2} \varnothing} = \frac{\sin \varnothing}{\cos \varnothing} \cdot \frac{\cos^{2} \varnothing}{\sin^{2} \varnothing} \] This simplifies to: \[ \frac{\sin \varnothing \cdot \cos^{2} \varnothing}{\sin^{2} \varnothing \cdot \cos \varnothing} = \frac{\cos \varnothing}{\sin \varnothing} = \cot \varnothing \] Thus, \( \tan \varnothing \cot^{2} \varnothing = \cot \varnothing \) is verified as true.