Verify that the trigonometric equation is an identity. \[ \frac{\csc ^{2} t+1}{\cot ^{2} t}=\frac{\csc t+\sin t}{\csc t-\sin t} \] Which of the following statements establishes the identity? A. \[ \frac{\csc ^{2} t+1}{\cot ^{2} t}=\frac{\csc ^{2} t-1}{\csc ^{2} t+1}=\frac{\csc t+\sin t}{\csc t-\sin t} \] B. \[ \frac{\csc ^{2} t+1}{\cot ^{2} t}=\frac{\csc ^{2} t+1}{\csc ^{2} t-1}=\frac{\csc t+\sin t}{\csc t-\sin t} \] c. \[ \frac{\csc ^{2} t+1}{\cot ^{2} t}=\frac{\sec ^{2} t+1}{\sec ^{2} t-1}=\frac{\csc t+\sin t}{\csc t-\sin t} \] D. \[ \frac{\csc ^{2} t+1}{\cot ^{2} t}=\frac{\sec ^{2} t-1}{\sec ^{2} t+1}=\frac{\csc t+\sin t}{\csc t-\sin t} \]

Step 1 ：The given trigonometric equation is an identity. To verify this, we need to simplify both sides of the equation and check if they are equal.

Step 2 ：We know that \(\csc^2 t = 1 + \cot^2 t\) and \(\csc t = 1/\sin t\).

Step 3 ：We can substitute these values into the equation and simplify.

Step 4 ：\(t = t\)

Step 5 ：lhs = \((\csc(t)^2 + 1)\sin(t)^2\)

Step 6 ：rhs = \((\sin(t) + \csc(t))/(-\sin(t) + \csc(t))\)

Step 7 ：simplified_lhs = \sin(t)^2 + 1

Step 8 ：simplified_rhs = -1 + 2/\cos(t)^2

Step 9 ：The simplified left-hand side (LHS) and right-hand side (RHS) of the equation are not equal. This means that the original equation is not an identity.

Step 10 ：However, the question asks us to verify which of the given statements establishes the identity. We need to check each of the given options and see if any of them are equal to the original equation.

Step 11 ：After simplifying the left-hand side (LHS) and right-hand side (RHS) of each option, we can see that only Option B gives us the same result for both sides.

Step 12 ：Therefore, Option B is the correct answer.

Step 13 ：Final Answer: The correct statement that establishes the identity is \(\boxed{\text{Option B}}\).