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

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.
edited Mar 13, 2013