HELP establish Trig Identity: 1-2cos^2 theta/ sin theta cos theta...?

Start with the left side until it equals the right side. Please show ALL work/steps, my algebra is a little rusty and I need to remember what I can and can't do. thanks!

1-2cos^2 theta/ sin theta cos theta = tan theta - cot theta

or written to see easier:

1 - 2cos^2(x) / sin(x) cos(x) = tan(x) - cot(x)

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  • 7 years ago
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    The main part of your algebra that needs repair has to do with

    "order of operations" - look this up in Wikipedia.

    The identity is FALSE the way you typed it,

    but what you meant was

    [1 - 2 cos^2(x)] / [sin(x) cos(x)] = tan(x) - cot(x)

    Since 1 = sin^2(x) + cos^2(x), the following is equivalent to the line above:

    [ sin^2(x) - cos^2(x) ] / [sin(x) cos(x)] = tan(x) - cot(x)

    Then separate the two terms of the numerator:

    sin^2(x)/[sin(x)cos(x)] - cos^2(x)/[sin(x)cot(x)] = tan(x) - cot(x)

    and reduce the fractions:

    sin(x)/cos(x) - cos(x)/sin(x) = tan(x) - cot(x)

    But now it's perfectly clear that the LHS equals the RHS.

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