(1)

RBC.ID.1: Trigonometric Identity

(2)

proof

(3)

(4)

by geometric series

(5)

replacing (5) by,

(6)

results

(7)

guided by the following identities:

(8)

(9)

(10)

from (7) , (8) , (9) , and (10) , The first part of (7):

(11)

The first part of (11) equals zero

(12)

(13)

(14)

from (7) , (8) , (9) , and (10) , The second part of (7):

(15)

(16)

(17)

(18)

from (14 and (18) ,

(19)

if x=1/2 , then

(20)

RBC.ID.2: Derivative of Trigonometric Identity-1

(21)

Proof

differntiate (19) in respect to x

(22)

(23)

(24)

multiply and divide the first part of right-hand side by

(25)

(26)

from (25) and (26)

(27)

(28)

if x=1/2

(29)

divide both sides by 4

(30)

RBC.ID.3: Derivative of Trigonometric Identity-2

Proof

(31)

(32)

(33)

Using geometric mean

(34)

(35)

Replacing

(36)

(37)

(38)

From (31) and (38) , Therefore ,

(39)