SOLUTION: Given: sin(x) - cos(x) = 1/(5^1/2) find the value of tan(x) + cot(x) + cos(2x)

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Question 1199966: Given: sin(x) - cos(x) = 1/(5^1/2) find the value of tan(x) + cot(x) + cos(2x)
Answer by ikleyn(52814) About Me  (Show Source):
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Given: sin(x) - cos(x) = 1/(5^1/2) find the value of tan(x) + cot(x) + cos(2x)
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The given equality  sin(x) - cos(x) = 1%2Fsqrt%285%29  implies after squaring


    sin^2(x) - 2sin(x)*cos(x) + cos^2(x) = 1%2F5

    1 - sin(2x) = 1%2F5

    1 - 1%2F5 = sin(2x)

    sin(2x) = 4%2F5.      (1)


The value of  tan(x) + cot(x) + cos(2x) is

      sin%28x%29%2Fcos%28x%29 + cos%28x%29%2Fsin%28x%29 + cos(2x) = 

    = %28sin%5E2%28x%29+%2B+cos%5E2%28x%29%29%2F%28sin%28x%29%2Acos%28x%29%29 + cos(2x) = 

    = 2%2Fsin%282x%29 + cos(2x).


Substitute here sin(2x) = 4%2F5  and  cos(2x) = sqrt%281-sin%5E2%282x%29%29 = sqrt%281-%284%2F5%29%5E2%29 = 3%2F5,  based on (1).  You will get

    tan(x) + cot(x) + cos(2x) = 2%2F%28%284%2F5%29%29 + 3%2F5 = 10%2F4 + 3%2F5 = %285%2A10%29%2F20 + %284%2A3%29%2F20 = 

                              = %2850%2B12%29%2F20 = 62%2F20 = 31%2F10 = 3.1.    ANSWER


ANSWER.  If  sin(x) - cos(x) = 1%2Fsqrt%285%29  then  tan(x) + cot(x) + cos(2x) = 31%2F10 = 3.1.

Solved.