Trigonometry

posted by Fern Angel

Prove the following identities:
1. (tan theta - sin theta)^2 + (1-cos theta)^2 = (1-sec theta) ^2
2. (1-2cos^2 theta) / sin theta cos theta = tan theta - cot theta
3. (sin theta + cos theta ) ^2 + (sin theta - cos theta ) ^2 = 2


Thank you so much! :)

  1. Steve

    sinθ = tanθ cosθ, so
    tanθ - sinθ = tanθ (1-cosθ)
    squared that is tan^2θ (1-cosθ)^2

    now we can add up the left side:

    tan^2θ(1-cosθ)^2 + (1-cosθ)^2
    = (tan^2θ + 1)(1-cosθ)^2
    = sec^2θ (1-cosθ)^2

    on the right side, we have

    1-secθ = (cosθ-1)/cosθ
    squared, that is sec^2θ (cosθ-1)^2

    and they are the same

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