Which from the following is true? (A) $\cos \theta=\left[\left(\left|\mathrm{A}^{\rightarrow} \times \mathrm{B}^{\rightarrow}\right|\right) / \mathrm{AB}\right]$ (B) $\sin \theta=\left[\left(\mathrm{A}^{\rightarrow} \cdot \mathrm{B}^{\rightarrow}\right) / \mathrm{AB}\right]$ (C) $\tan \theta=\left[\left(\left|\mathrm{A}^{\rightarrow} \times \mathrm{B}^{\rightarrow}\right|\right) /\left(\mathrm{A}^{-}-\mathrm{B}^{-}\right)\right]$ (D) $\cot \theta=\left[\mathrm{AB} /\left(\left|\mathrm{A}^{\rightarrow} \times \mathrm{B}^{\rightarrow}\right|\right)\right]$

The dot product A⃗.B⃗ of two vectors can be represented as \(|\mathrm{A}^{\rightarrow}| |\mathrm{B}^{\rightarrow}| \cos \theta\), where θ is the angle between A⃗ and B⃗. Rearranging the formula, we get \(\cos \theta = \frac{\mathrm{A}^{\rightarrow} \cdot \mathrm{B}^{\rightarrow}}{|\mathrm{A}^{\rightarrow}||\mathrm{B}^{\rightarrow}|}\).

The cross product |\(A⃗ \times B⃗\)| can be represented as \( |\mathrm{A}^{\rightarrow}||\mathrm{B}^{\rightarrow}| \sin \theta\), where θ is the angle between A⃗ and B⃗. Rearranging the formula, we get \(\sin \theta = \frac{|\mathrm{A}^{\rightarrow} \times \mathrm{B}^{\rightarrow}|}{|\mathrm{A}^{\rightarrow}||\mathrm{B}^{\rightarrow}|}\).

(A) The statement is incorrect because it confuses the cross product and cosine relationship. The correct equation for cosine is: \(\cos \theta = \frac{\mathrm{A}^{\rightarrow} \cdot \mathrm{B}^{\rightarrow}}{|\mathrm{A}^{\rightarrow}||\mathrm{B}^{\rightarrow}|}\). (B) The statement is incorrect because it confuses the dot product and sine relationship. The correct equation for sine is: \(\sin \theta = \frac{|\mathrm{A}^{\rightarrow} \times \mathrm{B}^{\rightarrow}|}{|\mathrm{A}^{\rightarrow}||\mathrm{B}^{\rightarrow}|}\). (C) The statement is incorrect because the denominator is not part of the tangent relationship and does not include any valid vector operations. (D) The statement is correct. Recalling the sine relationship from earlier, we have \(\sin \theta = \frac{|\mathrm{A}^{\rightarrow} \times \mathrm{B}^{\rightarrow}|}{|\mathrm{A}^{\rightarrow}||\mathrm{B}^{\rightarrow}|}\). Inversely, \(\csc \theta = \frac{|\mathrm{A}^{\rightarrow}||\mathrm{B}^{\rightarrow}|}{|\mathrm{A}^{\rightarrow} \times \mathrm{B}^{\rightarrow}|}\). The cotangent function is the reciprocal of the tangent function, and since \(\cot \theta = \frac{ \cos \theta}{\sin \theta}\) and \(\csc \theta = \frac{1}{\sin \theta}\), we can find that \(\cot \theta=\frac{\mathrm{AB}}{\left|\mathrm{A}^{\rightarrow} \times \mathrm{B}^{\rightarrow}\right|}\), which matches the given statement D.