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Math Help - help with proving trig indentity

  1. #1
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    Question help with proving trig indentity

    I need to prove that cosx+cotx/secx+tanx = cosxcotx, any help would be greatly appreciated!
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  2. #2
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    Re: help with proving trig indentity

    $\cos(x)+\dfrac{\cot(x)}{\sec(x)}+\tan(x) \overset{?}{=} \cos(x)\cot(x)$

    $\cos(x)+\dfrac{\dfrac{\cos(x)}{\sin(x)}}{\dfrac{1 }{\cos(x)}}+\dfrac{\sin(x)}{\cos(x)} \overset{?}{=} \cos(x)\dfrac{\cos(x)}{\sin(x)}$

    $\cos(x)+\dfrac{\cos^2(x)}{\sin(x)} + \dfrac{\sin(x)}{\cos(x)} \overset{?}{=} \dfrac{\cos^2(x)}{\sin(x)}$

    $\cos(x) + \dfrac{\sin(x)}{\cos(x)} \overset{?}{=} 0$

    $\cos^2(x) \overset{?}{=} -\sin(x)$

    No. These two expressions are not equal.

    If you meant

    $\cos(x)+\dfrac{\cot(x)}{\sec(x)+\tan(x)} \overset{?}{=} \cos(x)\cot(x)$

    Then I suggest you learn to use parentheses and use the method I followed above.
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  3. #3
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    Re: help with proving trig indentity

    Hello, cricket71!

    \text{Prove: }\:\frac{\cos x+\cot x}{\sec x+\tan x} \:=\: \cos x\cot x

    We have: . \frac{\cos x+\cot x}{\sec x+\tan x}

    Multiply by \tfrac{\sec x - \tan x}{\sec x - \tan x}

    . . \frac{\cos x+\cot x}{\sec x+\tan x} \cdot \frac{\sec x - \tan x}{\sec x - \tan x}

    . . =\;\frac{\overbrace{\cos x\sec x}^{1} - \overbrace{\cos x\tan x}^{\sin x} +\overbrace{\cot x\sec x}^{\frac{1}{\sin x}} - \overbrace{\cot x\tan x}^1}{\underbrace{\sec^2\!x - \tan^2\!x}_1}

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


    . . =\;\cos x\cdot\frac{\cos x}{\sin x} \;=\;\cos x\cot x
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