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Math Help - Help Verifying an Identity

  1. #1
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    Help Verifying an Identity

    Hi all!

    I got myself stuck here trying to figure out how to verify this Identity.


    tan x + cot x = sec x csc x

    Here is what I have so far

    tan x + cot x

    1/cot x + cot x/1

    (1+cot^2)/cot x

    (csc x/cot x) * csc x

    (1/sin x)/(1/tan x) * csc x

    (tan x/sin x) * csc x


    But I can't seem to turn (tan x/sin x) into sec x
    thankx for helping

    ps i'm still new to this so please be as detailed with your responses as you can thanks.
    Last edited by mathgex; December 1st 2011 at 04:24 PM.
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  2. #2
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    Re: Help Verifying an Identity

    tan x + cot x = (sin^{2}x + cos^{2}x)/(sin x cos x) = 1 /(sin x cos x) = sec x csc x
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  3. #3
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    Re: Help Verifying an Identity

    What steps did you take to get from

    tan x + cot x

    to

    (sin^{2}x + cos^{2}x)/(sin x cos x) ?
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  4. #4
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    Re: Help Verifying an Identity

    tan x + cot x = tan x * (sin x / sin x) + cot x * (cos x / cos x) = (sin x / cos x) * (sin x / sin x) + (cos x / sin x)* (cos x / cos x)
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  5. #5
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    Re: Help Verifying an Identity

    thank you very much i was able to follow along and see how it was solved.. very helpful
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  6. #6
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    Re: Help Verifying an Identity

    I have another identity I am having trouble with if any1 can help..

    (sin x/1-cot x) - (cos x/tan x-1) = sin x + cos x

    I was able to clear the denominators but then I am not sure how to go about making the sign positive.

    [(sin x/1-cot x)*(tan x-1/tan x-1)] - [(cos x/tan x-1)*(1-cot x/1-cot x)]

    = sin x - cos x

    thanx
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  7. #7
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    Re: Help Verifying an Identity

    Hello, mathgex!

    \frac{\sin x}{1-\cot x} - \frac{\cos x}{\tan x-1} \:=\: \sin x + \cos x

    We have: . \frac{\sin x}{1 - \frac{\cos x}{\sin x}} - \frac{\cos x}{\frac{\sin x}{\cos x} - 1}


    Multiply the first fraction by \frac{\sin x}{\sin x}, the second fraction by \frac{\cos x}{\cos x}

    . . \frac{\sin x}{\sin x}\cdot\frac{\sin x}{1 - \frac{\cos x}{\sin x}} \:-\: \frac{\cos x}{\cos x}\cdot\frac{\cos x}{\frac{\sin x}{\cos x} - 1}

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


    . . =\;\frac{(\sin x - \cos x)(\sin x + \cos x)}{\sin x - \cos x} \;=\;\sin x + \cos x

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  8. #8
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    Re: Help Verifying an Identity

    thanks soroban,

    it took me a second to realize the difference of two squares move you made
    thank you for helping, whats a good step by step method for solving these things?
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