The correct statement that establishes the identity is the third one: "The reciprocal of the cosine of an angle is equal to the secant of the angle."
Here's how you can prove this using the other three statements:
"The tangent of an angle is equal to the sine of the angle divided by the cosine of the angle."
"The reciprocal of the sine of an angle is equal to the cosecant of the angle."
"The reciprocal of the cosine of an angle is equal to the secant of the angle."
"The reciprocal of the tangent of an angle is equal to the cotangent of the angle."
To prove the identity, we can start by substituting the first statement into the equation on the left side of the identity:
$1 + \tan^2 (-0) = 1 + \left( \frac{\sin (-0)}{\cos (-0)} \right)^2 = 1 + \frac{\sin^2 (-0)}{\cos^2 (-0)}$
Next, we can use the fourth statement to write the right side of the identity in terms of the cotangent:
$1 + \frac{\sin^2 (-0)}{\cos^2 (-0)} = 1 + \frac{1}{\cot^2 (-0)} = \frac{1}{\cot^2 (-0)}$
Now we can use the second statement to write the right side of the identity in terms of the cosecant:
$\frac{1}{\cot^2 (-0)} = \frac{1}{\frac{1}{\sin^2 (-0)}} = \csc^2 (-0)$
Finally, we can use the third statement to write the right side of the identity in terms of the secant:
$\csc^2 (-0) = \frac{1}{\sec^2 (-0)} = \sec^2 (-0)$
Therefore, the third statement establishes the identity.