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Prove: $\cos^{-1}x-\cos^{-1}y=\cos^{-1}\bigg[xy+\sqrt{1-x^2}.\sqrt{1-y^2}\bigg]$

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Toolbox:
  • \( cos(A-B) = cosA\: cosB+ sinA\: sinB\)
  • \(1-cos^2A=sin^2A\)
Let \( cos^{-1}x=A\: and \: cos^{-1}y=B\)
\( \Rightarrow x=cosA \: and \: y=cos B\)
\(xy+\sqrt{1-x^2}\: \sqrt{1-y^2}=\)
\(cosAcosB+\sqrt{1-cos^2A}\sqrt{1-cos^2B}\)
\( = cosA\: cos B+ sinA\: sinB\)
\( =cos(A-B)\) (from the above formula)
Substituting the values of x and y in R.H.S
R.H.S \( = cos^{-1}(xy+\sqrt{1-x^2}\: \sqrt{1-y^2}) \)
\( cos^{-1}\: cos (A-B)\)
\( = A-B \)
\( = cos^{-1}x-cos^{-1}y\)
=L.H.S.
answered Feb 19, 2013 by thanvigandhi_1
edited Mar 13, 2013 by rvidyagovindarajan_1
 

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