Advanced Math

Could someone help me with those two questions. I need to know how to do them without calculator

solve for the interval [0,2pie]

1) cos^2*x-cos2x=0

2) tan(x/2)-sinx=0

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  1. 1.
    cos^sup2;(x)-cos(2x) = 0
    (1-sin²(x)) - (cos²(x)-sin²(x)) = 0
    (1-sin²(x)) - cos²(x) + sin²(x) = 0
    1-cos²(x) = 0
    sin²(x) = 0
    On the interval [0,2π],
    sin²(x) = 0 at x=0, π and 2π

    Subsititute x=0, π and 2π into the original equation to check each of the solutions.

    a. 2 trivial solutions
    By inspection, at x=0 and x=2π,

    b. other solutions
    tan(x/2)=(1-cos(x))/sin(x), and
    1-sin²(x) = cos²(x)
    to simplify the expression to:
    cos(x)(cos(x)-1)/sin(x) = 0
    or cos(x)=0, or cos(x)=1
    to give x=π/2, 3π/2, 0, 2π
    Check each solution to confirm its validity.

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    posted by MathMate

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