Prove that: `2cos(pi/13)cos({9pi}/13)+cos({3pi}/13)+cos({5pi}/13)=0` 

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lfryerda's profile pic

lfryerda | High School Teacher | (Level 2) Educator

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To prove that this equation is always true, we use the trigonometric product-to-sum identity (given in the website below):

`2cos({a-b}/2)cos({a+b}/2)=cosa+cosb`

Letting `a={5pi}/13` and `b={3pi}/13` , we get:

`LS=2cos(pi/13)cos({9pi}/13)+cos({3pi}/13)+cos({5pi}/13)`

`=2cos(pi/13)cos({9pi}/13)+2cos(pi/13)cos({4pi}/13)`

`=2cos(pi/13)(cos({9pi}/13)+cos({4pi}/13))`  now use the identity again

`=2cos(pi/13)2cos({5pi}/26)cos(pi/2)`   but `cos(pi/2)=0`

`=0=RS`

The equation is proven true.

Sources:
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vaaruni | High School Teacher | (Level 1) Salutatorian

Posted on

Require to prove :

         2 cos(pi/13) cos(9pi/13) + cos(3pi/13)+cos(5 pi/13) = 0

   Let us take  L.H.S

   L.H.S=> 2 cos(pi/13) cos(9 pi/13) + cos(3 pi/13)+cos(5 pi/13)

     => 2 cos(pi/13) cos(9 pi/13) + cos(3 pi/13)+cos(5 pi/13) 

=>2cos(pi/13)cos(9 pi/13)+2cos((3 pi+5 pi)/2)/13)cos((5pi-3pi/13)

[ By applying : cosA+ cosB = 2cos((A+B)/2) cos((A-B)/2) ]

 =>2 cos(pi/13) cos(9 pi/13) + 2 cos((8 pi/2)/13) cos((2 pi/2)/13) 

   => 2 cos(pi/13) cos(9 pi/13) + 2cos(4 pi/13) cos(pi/13)

[Since, (3 pi +5 pi)/2=8 pi /2 = 4 pi  &  (5 pi – 3 pi)/2 = 2 pi/2= pi ]

   => 2 cos(pi/13) cos(9 pi/13) + 2 cos(pi- (9pi/13)) cos(pi/13)

                [ Since,  cos(4 pi/13) = cos(pi- 9 pi/13)  ]  

   => 2 cos(pi/13) cos(9 pi/13)  - 2 cos(pi/13) cos(9 pi/13)

             [ since cos(180-A) = - cosA]

    =>  0 =  R.H.S

            L.H.S  =  R.H.S   

 Hence,  

2 cos(pi/13) cos(9pi/13) + cos(3pi/13)+cos(9pi/13) = 0  Proved   

 

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