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Math Help - Verifying Identities

  1. #1
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    Verifying Identities

    1. a. Use a calculator to determine which one of the following equations is an identity.



    1/ sin x + cos x = csc x + sec x




    1/ 1 - sin^2 x = 1 + tan^2x




    b. Verify the identity equation.




    2. Verify the following identity.

    tan x-tan y = sin(x-y) / cos x cos y
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  2. #2
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    just give a value to x and see if both sides are equal..
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  3. #3
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by accordry View Post
    1. a. Use a calculator to determine which one of the following equations is an identity.



    1/ sin x + cos x = csc x + sec x




    1/ 1 - sin^2 x = 1 + tan^2x




    b. Verify the identity equation.
    Graph these in your calculator and see if the graphs are the same. If not, then the identities are false.


    2. Verify the following identity.

    tan x-tan y = sin(x-y) / cos x cos y
    \tan x=\frac{\sin x}{\cos x} and \tan y=\frac{\sin y}{\cos y}.

    Thus, \tan x-\tan y=\frac{\sin x}{\cos x}-\frac{\sin y}{\cos y}

    Now find a common denominator. In this case, it would be \cos x\cos y

    Thus, \frac{\sin x}{\cos x}-\frac{\sin y}{\cos y}=\frac{\sin x}{\cos x}\cdot\frac{\cos y}{\cos y}-\frac{\sin y}{\cos y}\cdot\frac{\cos x}{\cos x}=\frac{\sin x\cos y-\cos x\sin y}{\cos x\cos y}

    Note that \sin x\cos y-\cos x\sin y=\sin(x-y)

    Thus, \frac{\sin x\cos y-\cos x\sin y}{\cos x\cos y}=\frac{\sin(x-y)}{\cos x\cos y}

    We have established the identity.

    Does this make sense?

    --Chris
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  4. #4
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    Hello, accordry!

    2. Verify: . \tan x - \tan y \:=\: \frac{\sin(x-y)}{\cos x\cos y}
    On the right side, we have:

    \frac{\sin(x-y)}{\sin x\cos y} \;=\;\frac{\sin x\cos y - \cos x\sin y}{\cos x\cos y} \;=\;\frac{\sin x\,{\color{red}\rlap{/////}}\cos y}{\cos x\,{\color{red}\rlap{/////}}\,\cos y} - \frac{{\color{red}\rlap{/////}}\,\cos x\sin y}{{\color{red}\rlap{/////}}\cos x\,\cos y}

    . . . . . = \;\frac{\sin x}{\cos x} - \frac{\sin y}{\cos y} \;=\;\tan x - \tan y

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  5. #5
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    how would you put the 1st ones in your calculator
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