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Trig Identities

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Please help...I'm not understanding trig identities and how to manipulate and express these two problems in their associated functions.

Thanks

a)Express as a function of cos (theta)

2 sin^2(theta) - 1


b)Express as a function of sin (theta) or cos (theta)

tan^2(theta) - 2 sec(theta)sin(theta)

  • Trig Identities -

    I will use x instead of theta for easier typing

    2 sin^2 x - 1 = cos 2x
    do you have to prove this?
    if so, then

    RS = -cos(2x)
    = -cos(x+x)
    = -cosxcosx + sinxsinx
    = -cos^x + sin^x
    = -(1-sin^x_ + sin^x
    = 2sin^x - 1
    = LS

  • Trig Identities -

    for the second

    tan^x - 2secxsinx
    = sin^2 x/cos^2 x - 2sinx/cosx
    = sin^2 x/cos^2 x 2 sinxcosx/cos^2 x
    = (sin^2 x - sin 2x)/(1 - sin^2 x)

    this is in terms of sinx

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