If $A=\int_1^{\sin \theta} \frac{t}{1+t^2} d t$ and $B=\int_1^{\operatorname{cosec} \theta} \frac{1}{t\left(1+t^2\right)} d t$ then the value of $\left|\begin{array}{ccc}A & A^2 & B \\ e^{A+B} & B^2 & -1 \\ 1 & A^2+B^2 & -1\end{array}\right|=$

• A

  $\sin \theta$
• B

  $\operatorname{cosec} \theta$
• Correct answer
  C

  0
• D

  1

Hence, the correct answer is option: 'C'.