1. **Problem statement:** Given a circle with points A, B, C, and F on the circumference, and a tangent DE touching the circle at point B. The angle at F is 96° and the angle between the tangent DE and line BC at point B is 36°. We need to find the value of $\angle BCA$. 2. **Key theorem:** The angle between a tangent and a chord through the point of contact equals the angle in the alternate segment of the circle. This means: $$\angle DBE = \angle BCA$$ where $\angle DBE$ is the angle between tangent DE and chord BC at B. 3. From the problem, $\angle DBE = 36^\circ$. 4. Therefore, by the alternate segment theorem: $$\angle BCA = 36^\circ$$ 5. The angle at F (96°) is extra information and does not affect the calculation of $\angle BCA$ in this context. **Final answer:** $$\boxed{36^\circ}$$