Conversion from infinite product to continued fraction

From an obvious rule,

  in special form

For example

Applying because

 while ,

 from the rule of signs(up->front)

From the rule

as in 0128,

　by rules of signs

Apply gamma-shaped rules repeatedly, we have

From one of the rules in 0113,

Applying gamma-shaped multipliers, we finally have

For sin(x) for example