Establish the following identity. \( \begin{array}{l}\frac{1-\cos \theta}{1+\cos \theta}=\frac{\sec \theta-1}{\sec \theta+1} \\ \text { To establish the identity, start with the side containing the more complicated expression. Since both sides have simple expressions, start with either of the expressions. Here, start with sec } \frac{\sec \theta-1}{\sec \theta+1} \sec \theta+1\end{array}=\frac{\square-1}{\square+1} \)

To establish the identity: \[ \frac{1 - \cos \theta}{1 + \cos \theta} = \frac{\sec \theta - 1}{\sec \theta + 1} \] we can start with the right-hand side (RHS) and simplify it to match the left-hand side (LHS). **Step 1:** Start with the RHS. \[ \frac{\sec \theta - 1}{\sec \theta + 1} \] **Step 2:** Express \(\sec \theta\) in terms of \(\cos \theta\). \[ \sec \theta = \frac{1}{\cos \theta} \] Substitute this into the RHS: \[ \frac{\frac{1}{\cos \theta} - 1}{\frac{1}{\cos \theta} + 1} \] **Step 3:** Combine the terms in the numerator and the denominator. \[ \frac{\frac{1 - \cos \theta}{\cos \theta}}{\frac{1 + \cos \theta}{\cos \theta}} \] **Step 4:** Simplify by multiplying the numerator and the denominator by \(\cos \theta\). \[ \frac{1 - \cos \theta}{1 + \cos \theta} \] **Conclusion:** After simplification, the RHS becomes: \[ \frac{1 - \cos \theta}{1 + \cos \theta} \] which matches the LHS. Therefore, the identity is established. \[ \frac{\sec \theta - 1}{\sec \theta + 1} = \frac{1 - \cos \theta}{1 + \cos \theta} \] **Final Answer:** \[ \frac{\sec \theta - 1}{\sec \theta + 1} = \frac{1 - \cos \theta}{1 + \cos \theta} \]