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Math Help - trigo

  1. #1
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    trigo

    Given that sin(x+y) = pcos(x-y) , show that

    tan x = (p-tan y)/(1-p tany)
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  2. #2
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    Hello, thereddevils!

    Given that: . \sin(x+y) \:=\: p\cos(x-y)

    show that: . \tan x \:=\:\frac{p-\tan y}{1-p\tan y}

    We have: . \sin(x+y) \;=\;p\cos(x-y)

    Then: . \sin x\cos y + \cos x\sin y \;=\;p(\cos x\cos y + \sin x\sin y)

    Rearrange terms: . \sin x\cos y - p\sin x\sin y \;=\;p\cos x\cos y - \cos x\sin y

    Factor: . \sin x(\cos y - p\sin y) \;=\;\cos x(p\cos y - \sin y)

    . . Then: . \frac{\sin x}{\cos x} \;=\;\frac{p\cos y - \sin y}{\cos y - p\sin y}


    On the right, divide numerator and denominator by \cos y:

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


    And we have: . \tan x \;=\;\frac{p-\tan y}{1 - p\tan y}

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