SOLUTION: If 8cosθ - 8sinθ = 3, find 55tanθ + 55/tanθ

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Question 1209754: If 8cosθ - 8sinθ = 3,
find 55tanθ + 55/tanθ
Answer by ikleyn(52752) About Me  (Show Source):
You can put this solution on YOUR website!
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If 8cos(a) - 8sin(a) = 3, find 55tan(a) + 55/tan(a)
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I will write letter 'a' everywhere instead of letter theta'.
It is because my text editor is not quite perfect. 


We are given 

    8cos(a) - 8sin(a) = 3,

     cos(a) -   sin(a) = 3%2F8.


Square both sides

     cos^2(a) - 2cos(a)*sin(a) + cos^2(a) = 9%2F64.


Replace here  cos^2(a) + sin^2(a) by 1

     1 - 2cos(θ)*sin(a) = 9%2F64,

     1 - 9%2F64 = 2cos(a)*sin(a),

      sin(2a) = 55%2F64.    (1)


From the other side hand,

    tan(a) + 1%2Ftan%28a%29 = %28tan%5E2%28a%29%2B1%29%2Ftan%28a%29 = %28%28sin%5E2%28a%29%2Fcos%5E2%28a%29%2B1%29%29%2F%28%28sin%28a%29%2Fcos%28a%29%29%29 = 

                            = %28%281%2Fcos%5E2%28a%29%29%29%2F%28%28sin%28a%29%2Fcos%28a%29%29%29 = 1%2F%28cos%28a%29%2Asin%28a%29%29 = 2%2Fsin%282a%29.   (2)

           
Therefore, from (1) and (2) we have

    55tan(a) + 55%2Ftan%28a%29 = %282%2A55%29%2Fsin%282a%29 = 110%2F%28%2855%2F64%29%29 = 2*64 = 128.    ANSWER

Solved.