Product of Multiple Trignometry Function

PMTF.ID.1:   Product of multiple sine

Proof

(1)

(2)

(3)

(4)

from (3) and (4) ,

(5)

(6)

from (6) and (5)

(7)

(8)

 

(9)

(10)

(11)

(12)

the equation (12) is represented as product of zeros , shown as following

(13)

(14)

from (12) and (14)

(15)

(16)

if x=1 , then

(17)

from (17) and (9)

(18)

(19)

PMTF.ID.1:   Product of multiple cosine

Proof

 

 (1)

 (2)

 

 (3)

 .
 .
 .
 .
  

 (4)

  (5)

 (6)

Multiple Sine Integration

MSI.ID.1:   Recurrence Relation

 (1)

 (2)

Proof

(3)

 

 (4)

(5)

(6)

(7)

(8)

(9)

 

 (10)

 

(11)

From (11) and (1)