SOLUTION: cos^3(A)×cos(3A)+sin^3(A)×sin(3A)=cos^3 (2A)

Algebra ->  Trigonometry-basics -> SOLUTION: cos^3(A)×cos(3A)+sin^3(A)×sin(3A)=cos^3 (2A)      Log On


   

Question 1068643: cos^3(A)×cos(3A)+sin^3(A)×sin(3A)=cos^3 (2A)
Answer by ikleyn(52786) About Me  (Show Source):
You can put this solution on YOUR website!
.
cos^3(A)×cos(3A)+sin^3(A)×sin(3A)=cos^3 (2A)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Unfortunately, the person who submitted this post didn't say what he/she wanted.
So, I will formulate it instead of him/her: "Prove an identity . . . " 

The keys are these two formulas:

    cos%5E3%28A%29  = %281%2F4%29%2Acos%283A%29+%2B+%283%2F4%29%2Acos%28A%29,        (1)   and

    sin%5E3%28A%29  = %281%2F4%29%2Asin%283A%29+-+%283%2F4%29%2Asin%28A%29.        (2)

(see the lesson Powers of trigonometric functions in this site).  When applying them, you will get

cos^3(A)*cos(3A) =  %281%2F4%29%2Acos%283A%29+%2B+%283%2F4%29%2Acos%28A%29%29%2Acos%283A%29,   (3)  and

sin^3(A)*sin(3A) = %28-1%2F4%29%2Asin%283A%29+-+%283%2F4%29%2Asin%28A%29%29%2Asin%283A%29.   (4)


So, adding and expanding (3) and (4), you will get

  cos%5E3%28A%29%2Acos%283A%29%2Bsin%5E3%28A%29%2Asin%283A%29 = 

=  = 

= [%281%2F4%29%2Acos%5E2%283A%29+-+%281%2F4%29%2Asin%5E2%283A%29] + [%283%2F4%29%2Acos%28A%29%2Acos%283A%29+%2B+%283%2F4%29%2Asin%28A%29%2Acos%283A%29%29] = 

    For the first  bracket  [ . . ]  apply the formula cos(2x) = . . .   
    For the second bracket  [ . . ]  apply the formula cos(x-y) = . . .   You will get

= %281%2F4%29%2Acos%286A%29+%2B+%283%2F4%29%2Acos%28A-3A%29 = %281%2F4%29%2Acos%286A%29+%2B+%283%2F4%29%2Acos%282A%29 = 

    Now apply again the formula (1). You will get

= cos%5E3%282A%29.

QED. Proved and solved.