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Thread: trig identities question

  1. #1
    Cadet
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    Post trig identities question

    Prove this question:

    sin ( θ + α ) sin ( θ − α ) = sin2 θ − sin2 α

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    Re: trig identities question

    Use the identity:

    HSC 2018: [English Adv.] • [Maths Ext. 1] • [Maths Ext. 2] • [Chemistry] • [Software Design]

    Goal: ≥92.00 for B Advanced Mathematics (Hons) / B Engineering (Hons) (Computer) at UNSW

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    Re: trig identities question

    Using sin(A+B) = (sin A) cos(B) + cos(A) sin(B) and sin(A-B) = (sin A) cos(B) - cos(A) sin(B)

    To prove sin ( θ + α ) sin ( θ − α ) = sin^2θ − sin^2a
    Taking the LHS,

    sin ( θ + α ) sin ( θ − α )
    = [sin(θ)cos(a)+cos(θ)sin(a)] [sin(θ)cos(a)-cos(θ)sin(θ)]
    =sin^2θcos^2a-cos^2θsin^2a (Expanding and cancelling)
    = sin^2θ(1-sin^2a)-(1-sin^2θ)sin^2a (Using sin^2θ+cos^θ = 1)
    =sin^2θ-sin^2θsin^2a-[sin^2a-sin^2θsin^2a)
    =sin^2θ-sin^2θsin^2a-sin^2a+sin^2θsin^2a)
    = sin^2θ - sin^2a=RHS

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