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Math Help - Verifying Trig Identities

  1. #1
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    Question Verifying Trig Identities

    1. Verify the identity: 2cos4xsin2x=2sin3xcos3x-2cosxsinx
    2. Verify the identity: (1/1-cosx)-(cosx/1+cosx)=2csc(squared)x-1
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  2. #2
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    Hello, kelsey3!

    2. Verify the identity: . \frac{1}{1-\cos x} - \frac{\cos x}{1+\cos x} \;=\;2\csc^2\!x-1
    On the left side, get a common denominator and combine . . .


    \frac{1}{1-\cos x}\cdot{\color{blue}\frac{1+\cos x}{1 + \cos x}} - \frac{\cos x}{1+\cos x}\cdot{\color{blue}\frac{1-\cos x}{1-\cos x}} \;\;=\;\;\frac{1+\cos x}{1-\cos^2\!x} - \frac{\cos x(1 - \cos x)}{1-\cos^2\!x}


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


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


    . . = \;\frac{2}{\sin^2\!x} - \frac{\sin^2\!x}{\sin^2\!x} \;\;=\;\;2\csc^2\!x - 1

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