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)

Central Binomial Coeffecient

(1)

CBD.ID.1:   Recurrence Relation

 

(2)

 

 (3)

 

(4)

where

(5)

CBD.ID.2:  Generating Function

 (6)

Differentiating both sides with respect to x

 

(7)

(8)

using the recurrence relation  (4)

 (9)

 

(10)

(11)

(12)

 

 (13)

integrating both sides with respect to x ,

(14)

(15)

since

therefore,

 

(16)

CBD.ID.3:   Trigonometric Identity

 

proof

replacement  in (16) by :

 

 

therefore

 

 (17)

 

(18)

from (17) and (18) , therefore, 

 

(19)

integrate (19) from 0 to t

(20)

therefore, 

(21)

CBD.ID.4:   Catalan Number  Identity

 

Proof

integrating (16) from 0 to t

(22)

replacing 1-4x by u in (23)

(23)

 

(24)

(25)

divide both sides by t

 (26)

(27)

(28)

CBD.ID.5:   Square  Identity of Central Binomial Coefficient

Proof

squaring both sides of (16)

(29)

applying geometric series (1) identity on the left side , and Cauchy product method  (2) on the right side;

 

(30)

equating the coefficients of    from both sides  , we get

(31)