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Math Help - Verify a trig identity

  1. #1
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    Verify a trig identity

    Hello, I'm doing an induction problem and in the basis step I have to verify this identity (assuming \sin{x/2}\neq 0):

    \sin{x}=\frac{\sin{(2x)}}{4\sin^2{\left(\frac{x}{2  } \right) }}-\frac{2\cos{\left( \frac{3x}{2} \right) }}{2\sin{\left( \frac{x}{2} \right) }}

    Getting a common denominator, I get the RHS to be

    \frac{\sin{(2x)}-4\sin{\left( \frac{x}{2} \right) }\cos{\left( \frac{3x}{2} \right) }}{4\sin^2{\left( \frac{x}{2} \right) }}

    But where do I go from here? I'm afraid I have almost no background in trigonometry, and so don't know the strategy for these types of problems.
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  2. #2
    MHF Contributor ebaines's Avatar
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    Re: Verify a trig identity

    There may be an easier way, but if you convert the right hand side to be all in terms of (x/2) it works out pretty easily. Start with:

     \sin (2x) = \sin (\frac x 2 + \frac {3x} 2) = \sin (\frac x 2 ) \cos (\frac {3x} 2) + \cos (\frac x 2 ) \sin (\frac {3x} 2 )

    then apply the identities:

     \cos (3a) = \cos^3(a) - 3 \sin^2 (a) \cos(a)

     \sin(3a) = 3 \cos^2(a) \sin (a) - \sin^3(a)

    using a = x/2.

    You should end up with the right hand side being:  2 \sin (\frac x 2) \cos ( \frac x 2), which is equivalent to  \sin (2 \times \frac x 2) = \sin(x)
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  3. #3
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    Re: Verify a trig identity

    Thanks a ton!
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  4. #4
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    Re: Verify a trig identity

    You can also consider the attached solution.
    Verify a trig identity-22-jun-13.png
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