Detailed explanation of the Γ reflection formula understandable by an AP Calculus student

Note: This is a description from N.N. Lebedev, Special Functions and Their Applications, Dover, New York, 1972, it is not my work but it can be used as starting point.

Lebedev uses in his section 1.2 (Some Relations Satisfied by the Gamma Function) a double-integral approach. From the well-known integral formula

$$\Gamma(z) = \int_{0}^{\infty}t^{z-1}e^{-t}\, d t \qquad (\Re z > 0)$$

temporarily assume $ 0 < \Re z < 1,\,$ use the formula for $1-z\,$ and get

$$\Gamma(z)\Gamma(1-z) = \int_{0}^{\infty}\int_{0}^{\infty}s^{-z}t^{z-1}e^{-(s+t)}\,ds\, dt \qquad (0<\Re z <1)$$

With the new variables $u = s + t, v = t/s$ this gives $$\Gamma(z)\Gamma(1-z) = \int_{0}^{\infty}\int_{0}^{\infty}\frac{v^{z-1}}{1+v}e^{-u}\,du\, dv = \int_{0}^{\infty}\frac{v^{z-1}}{1+v} dv = \frac{\pi}{\sin \pi z}$$

For the last step he refers to Titchmarsh. Then he uses continuation to extend the formula to all $z\in \mathbb{C}$ without the integers.