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Math Help - Verifying a half angle formula

  1. #1
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    Verifying a half angle formula

    Verifying a half angle formula-codecogseqn.gif

    I have tried this one several times and I feel like I am going in circles. Probably has an easy solution and I will be embarrassed. Any help or guidance will be appreciated. Thank You

    I used this to write the equation: Online LaTeX Equation Editor - create, integrate and download
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  2. #2
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    Re: Verifying a half angle formula

    On the RHS, write \tan x as \sin x/\cos x and \sec x as 1/\cos x, multiply top and bottom by \cos x and then, (assuming that you are allowed to), make use of the identities

    \sin 2A=\2\sin A\cos A

    and

    \cos 2A=2\cos^{2}A-1.
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  3. #3
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    Re: Verifying a half angle formula

    Hello, dudewithmathproblems!

     \text{Prove: }\:\tan\frac{x}{2} \;=\; \frac{\tan x}{\sec x + 1}

    We can work from the LHS . . .

    \tan\frac{x}{2} \;=\;\frac{\sin\frac{x}{2}}{\cos\frac{x}{2}} \;=\; \frac{\sqrt{\frac{1-\cos x}{2}}}{\sqrt{\frac{1+\cos x}{2}}} \;=\;\sqrt{\frac{1-\cos x}{1+\cos x}} \;=\;\sqrt{\frac{1-\cos x}{1+\cos x}\cdot\frac{1+\cos x}{1+\cos x}}

    . . . . . =\;\sqrt{\frac{1-\cos^2x}{(1+\cos x)^2}} \;=\; \sqrt{\frac{\sin^2x}{(1+\cos x)^2}} \;=\;\frac{\sin x}{1+\cos x}


    Divide numerator and denominator by \cos x\!:

    . . . . . \frac{\dfrac{\sin x}{\cos x}}{\dfrac{1}{\cos x} + \dfrac{\cos x}{\cos x}} \;=\;\frac{\tan x}{\sec x + 1}
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