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Math Help - show that arctanx = arcsin(x/sqrt (1+x^2)

  1. #1
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    show that arctanx = arcsin(x/sqrt (1+x^2)

    also prove that

    arccosx = 2arccos sqrt ( (x+1)/2 )

    for the first one i let the LHS be y but didnt know where to go next

    for the second one i cosined both sides but am still unsure
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  2. #2
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    y = \arctan (x)\; \Rightarrow \;\tan (y) = x\; \Rightarrow \;\sin (y) = ?
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  3. #3
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    Hello, b0mb3rz!

    Prove that: . \arccos x \:= \:2\arccos\left(\sqrt{\tfrac{x+1}{2}}\right)
    Let \theta \:=\:\arccos\left(\sqrt{\tfrac{x+1}{2}}\right)\qua  d \Rightarrow\quad \cos\theta \:=\:\sqrt{\tfrac{x+1}{2}} .[1]

    The right side is: . y \:=\:2\theta


    Take the cosine of both sides:

    . . \cos y \:=\:\cos(2\theta) \quad\Rightarrow\quad \cos y \:=\:2\cos^2\!\theta - 1 .[2]


    Substitute [1] into [2]: . \cos y \;=\;2\left(\sqrt{\tfrac{x+1}{2}}\right)^2-1 \;=\;x

    We have: . x \:=\:\cos y

    . . Then: .  \arccos x \:=\:\arccos(y) \:=\:\arccos(2\theta)


    Therefore: . \arccos x \;=\;2\arccos\left(\sqrt{\tfrac{x+1}{2}}\right)

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