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In summary, to prove that (d/dx)arcos(x) = -1/(Sqrt(1-x^2)), we can use the fact that cos is the inverse function of arccos and differentiate both sides. Another approach is to use trig identities to simplify the expression and think about a right triangle to find the value of the sine.
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alexparker
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Prove that (d/dx)arcos(x) = -1/(Sqrt(1-x^2)


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  • #2
cos is the inverse function of arccos. So cos(arccos(x))=x. Differentiate both sides. Did you try that?
 
  • #3
Or: if y= arccos(x) then x= cos(y). Differentiate that to find dx/dy= -sin(y) so that
dy/dx= -1/sin(y)= -1/sin(arccos(y)).

Use trig identities to simplify that (change sin(arccos(y)) to cos(arccos(y))=y) or think about a right triangle with "near side" of length y and hypotenuse of length 1. What is the length of the "opposite side"? Whatr is the sine?
 

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