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trig

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We are doing trig identities in school. I need help with these five:

1.1+sinx/1-sinx=cscx+1/cscx-1

2.tanx+sinx/1+cosx=tanx

3.sec^2x/sin^2x=1/sin^2x+1/cos^2x

4.tan^2x/1+tan^2x=1-cos^2x

5.sinx-sin^3x/cos^3x=tanx

  • trig -

    My method
    1. start with the more complicate looking side
    2. change everything to sines and cosines, unless I recognize an obvious identity

    I will do #2 for you

    LS = (tanx + sinx)/(1 + cosx)
    = (sinx/cosx + sinx)/(1+cosx)
    = [( sinx + sinxcosx)/cosx ]/(1+cosx)
    = [sinx(1+cosx)/cosx]/(1+cosx)
    = sinx/cosx
    = tanx
    = RS

    by "obvious identity" I notice in #4 that we have the expression
    1 + tan^2x
    from the Pythagorean identities
    1 + tan^2x = sec^2x

    so #4
    LS = tan^2x/(1+tan^2)
    = tan^2x/sec^2x
    = tan^2x cos^2x
    = sin^2x/cos^2x (cos^2x) = sin^2x
    = 1 - cos^2x
    = RS

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