Verify that the following equation is an identity. \[ \sin ^{2} \frac{\theta}{2}=\frac{\csc \theta-\cot \theta}{2 \csc \theta} \] Which of the following three statements verifies the given equation? A. $\sin ^{2} \frac{\theta}{2}=\left(\sqrt{\frac{1-\cos \theta}{2}}\right)^{2}=\frac{1-\cos \theta}{2} \cdot \frac{\csc \theta}{\csc \theta}=\frac{\csc \theta-\cot \theta}{2 \csc \theta}$ B. $\sin ^{2} \frac{\theta}{2}=\left(\sqrt{\frac{1-\cos \theta}{2}}\right)^{2}=\frac{1-\cos \theta}{2} \cdot \frac{\sec \theta}{\sec \theta}=\frac{\csc \theta-\cot \theta}{2 \csc \theta}$ c. $\sin ^{2} \frac{\theta}{2}=\left(\sqrt{\frac{1+\cos \theta}{2}}\right)^{2}=\frac{1+\cos \theta}{2} \cdot \frac{\csc \theta}{\csc \theta}=\frac{\csc \theta-\cot \theta}{2 \csc \theta}$
Solution
Step 1 ：The problem is asking to verify the given trigonometric identity. To do this, we need to use the trigonometric identities and properties. The given equation is in terms of sine, cosecant, and cotangent. We know that \(\sin ^{2} \frac{\theta}{2}=\frac{1-\cos \theta}{2}\), \(\csc \theta = \frac{1}{\sin \theta}\) and \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). We can use these identities to simplify the right side of the equation and see if it equals to the left side.
Step 2 ：Let's denote \(\theta\) as theta. The left side of the equation is \(\sin(\frac{\theta}{2})^2\).
Step 3 ：The right side of the equation is \((\frac{1}{\cos(\theta)} - \frac{\cos(\theta)}{\sin(\theta)})\cdot\frac{\cos(\theta)}{2}\).
Step 4 ：Simplify the right side of the equation, we get \(\frac{\sin(\theta)}{2} + \frac{1}{2} - \frac{1}{2\sin(\theta)}\).
Step 5 ：Comparing the left side and the right side of the equation, we can see that they are not equal, which means the given equation is not an identity.
Step 6 ：\(\boxed{\text{Final Answer: None of the statements A, B, or C verifies the given equation.}}\)
