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Math Help - 2 Special identities questions..

  1. #1
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    2 Special identities questions..

    a) Use the half-angle formulas for sine and cosine to verify the identity:
    tan(a/2) = (1-cosa)/(sina)

    b) Let x represent a real number. Find all solutions of the trigonometric equation:
    tan(x/2) + sinx = 0

    Explanations, steps would be helpful. Thank you!
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  2. #2
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    Quote Originally Posted by zuuberbat View Post
    a) Use the half-angle formulas for sine and cosine to verify the identity:
    tan(a/2) = (1-cosa)/(sina)

    b) Let x represent a real number. Find all solutions of the trigonometric equation:
    tan(x/2) + sinx = 0

    Explanations, steps would be helpful. Thank you!
    a) Use the double angle formulae:

    Substitute \cos a = \cos \left( 2 \cdot \frac{a}{2} \right) = 1 - 2\sin^2 \left(\frac{a}{2}\right) and \sin a = \sin \left( 2 \cdot \frac{a}{2} \right) = 2 \sin \left(\frac{a}{2}\right) \cos \left(\frac{a}{2}\right) into the right hand side and simplify.


    b) From a) the equation becomes

    \frac{1 - \cos x}{\sin x} + \sin x = 0

     \Rightarrow 1 - \cos x + \sin^2 x = 0

    \Rightarrow 1 - \cos x + (1 - \cos^2 x) = 0

    \cos^2 x + \cos x - 2 = 0

    \Rightarrow (\cos x + 2)(\cos x - 1) = 0

    The rest is left for you to do.

    NB: There's one small technical difficulty (left for you to spot) which you will need to justify.
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