How can I show whether the following function is Increasing, Decreasing, or Constant returns to scale?
You have $$F(x_1,x_2)=x_2\sin\bigg(\frac{x_1}{x_2}-\pi\bigg).$$
If you multiply all factors by $t$ you get
$$F(tx_1,tx_2)=t x_2\sin\bigg(\frac{tx_1}{tx_2}-\pi\bigg).$$
Simplify the fraction and you are almost done.