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Inverse sine and cosine

  1. Dec 12, 2007 #1
    To show that
    cos-1(-x)-cos-1(x)=2sin-1(x)

    I tried
    take x= sina
    taking cos of the whole equation
    cos(cos-1(-x))-cos(cos-1(x))=2cos(sin-1(x))
    now we have to prove : -x-x=2cos(sin-1(x))
    LHS: -2x=-2sina=2cos(a+pi/2)
    RHS: 2cosa

    Iam not sure how to proceed further..can anyone help me with this..
     
    Last edited: Dec 12, 2007
  2. jcsd
  3. Dec 12, 2007 #2

    marcusl

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    Science Advisor
    Gold Member

    This equation is incorrect. You need to expand cos(arcos(-x) - arcos(x)) properly,
    You will then need to use relations like cos(arsin(y))=sqrt(1-y^2)
     
  4. Dec 12, 2007 #3

    Dick

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    Homework Helper

    You could also proceed more concretely. Get out (or make) a graph of cos^(-1) and sin^(-1) (let's call them acos and asin). If cos(theta)=x then cos(pi-theta)=(-x). So acos(-x)-acos(x)=pi-2*theta. Now if cos(theta)=x then sin(pi/2-theta)=x. So asin(x)=???.
     
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