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    Question

    If 3 tan X + cot X = 2√3, then find the value of 6

    tan2 X + 2 cot2 X.
    A 6 Correct Answer Incorrect Answer
    B 3 Correct Answer Incorrect Answer
    C 5 Correct Answer Incorrect Answer
    D 8 Correct Answer Incorrect Answer

    Solution

    3 tan X + cot X = 2√3

    We know that, cot A = (1/tan A)

    3 tan X + (1/tan X) = 2√3

    Or, 3 tan2 X + 1 = 2√3 tan X

    Or, 3 tan2 X - 2√3 tan X + 1 = 0

    Or, 3 tan2 X - √3 tan X - √3tan X + 1 = 0

    Or, √3 tan X X (√3 tan X - 1) + ( - 1) X (√3 tan X - 1) = 0

    Or, (√3 tan X - 1) X (√3 tan X - 1) = 0

    Or, (√3 tan X - 1) 2 = 0

    Or, (√3 tan X - 1) = 0

    Or, tan X = (1/√3) = tan30o [because, tan30o = (1/√3) ]

    So, X = 30o
    6 tan2 X + 2 cot2 X = 6 tan2 30o + 2 cot2 30o

    = 6 X (1/√3) 2 + 2 X (√3) 2

    = 6 X (1/3) + 2 X 3 = 2 + 6 = 8

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