1. **State the problem:** Prove the identity $1 + \csc\theta = \csc\theta (1 + \sin\theta)$. 2. **Recall definitions and formulas:** Recall that $\csc\theta = \frac{1}{\sin\theta}$. This will help us rewrite the expressions in terms of sine. 3. **Evaluate the left side (LS):** $$LS = 1 + \csc\theta = 1 + \frac{1}{\sin\theta}$$ 4. **Evaluate the right side (RS):** $$RS = \csc\theta (1 + \sin\theta) = \frac{1}{\sin\theta} (1 + \sin\theta) = \frac{1}{\sin\theta} + \frac{\sin\theta}{\sin\theta} = \frac{1}{\sin\theta} + 1$$ 5. **Compare LS and RS:** $$LS = 1 + \frac{1}{\sin\theta} = \frac{1}{\sin\theta} + 1 = RS$$ 6. **Conclusion:** Both sides simplify to the same expression, so the identity is true. **Final answer:** The identity $1 + \csc\theta = \csc\theta (1 + \sin\theta)$ is verified.