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Math Help - Trigonometric Proof

  1. #1
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    Trigonometric Proof

    If cos x cos y = cos θ, prove that:


    sin^2 (x/2) cos^2 (y/2) +cos^2 (x/2) sin^2 (y/2) = sin^2 (θ/2)

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  2. #2
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    Quote Originally Posted by yobacul View Post
    If cos x cos y = cos θ, prove that:


    sin^2 (x/2) cos^2 (y/2) +cos^2 (x/2) sin^2 (y/2) = sin^2 (θ/2)

    Hint: Start with the formula \cos\phi = 1-2\sin^2(\phi/2). Apply that to all three angles x,\,y,\,\theta in the formula \cos x \cos y = cos \theta, and see if you rearrange the result to get the desired conclusion.
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  3. #3
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    Nope ...



    I tried to .... I managed to get what I required on the RHS but I didn't manage to rearrange the rest to get the LHS.
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  4. #4
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    Quote Originally Posted by yobacul View Post
    I managed to get what I required on the RHS but I didn't manage to rearrange the rest to get the LHS.
    \cos x\cos y = \cos\theta

    (1-2\sin^2(x/2))(1-2\sin^2(y/2)) = 1-2\sin^2(\theta/2)

    1 - 2\sin^2(x/2) -2\sin^2(y/2) + 4\sin^2(x/2)\sin^2(y/2) = 1-2\sin^2(\theta/2)

    Subtract 1 from both sides and divide by –2:

    \sin^2(x/2) +\sin^2(y/2) -2\sin^2(x/2)\sin^2(y/2) = \sin^2(\theta/2)

    Now write the left side of that as

    \sin^2(x/2)(1 - \sin^2(y/2)) + \sin^2(y/2)(1 - \sin^2(x/2))

    and you should be able to see how to complete the calculation from there.
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