Equation Verification for $\sin A = \cot B$

The problem requires finding the values of $A$ and $B$ that satisfy the trigonometric equation $\sin A = \cot B$. We will test the given options.

Checking Specific Values

Let's evaluate the equation $\sin A = \cot B$ for the values provided in the options.

• Option 1: $A = 0, B = 0$
  • $\sin A = \sin 0 = 0$
  • $\cot B = \cot 0$, which is undefined.
  • Therefore, this option is incorrect.
• Option 2: $A = \frac{\pi}{2}, B = \frac{\pi}{2}$
  • $\sin A = \sin \frac{\pi}{2} = 1$
  • $\cot B = \cot \frac{\pi}{2} = 0$
  • Since $1 \neq 0$, this option is incorrect.
• Option 3: $A = 0, B = \frac{\pi}{2}$
  • $\sin A = \sin 0 = 0$
  • $\cot B = \cot \frac{\pi}{2} = 0$
  • Since $0 = 0$, this option satisfies the equation.
• Option 4: $A = \frac{\pi}{2}, B = 0$
  • $\sin A = \sin \frac{\pi}{2} = 1$
  • $\cot B = \cot 0$, which is undefined.
  • Therefore, this option is incorrect.

Conclusion on Values

The only values that satisfy the equation $\sin A = \cot B$ among the choices are $A = 0$ and $B = \frac{\pi}{2}$.