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Question Number 10948 by rish@bh last updated on 03/Mar/17
If cos^(−1) (x/a)+cos^(−1) (y/b)=α  prove   (x^2 /a^2 )−((2xy)/(ab))cos α+(y^2 /b^2 )=sin^2 α
Ifcos1xa+cos1yb=αprovex2a22xyabcosα+y2b2=sin2α
Answered by ajfour last updated on 03/Mar/17
    let θ+∅=α  cos (θ+∅)=cos α  cos θcos ∅−sin θsin ∅=cos α  ((x/a))((y/b))−((√(a^2 −x^2 ))/a).((√(b^2 −y^2 ))/b)=cos α  (((xy)/(ab))−cos α)^2 =(1−(x^2 /a^2 ))(1−(y^2 /b^2 ))  −((2xycos α)/(ab))+cos^2 α =1−(x^2 /a^2 )−(y^2 /b^2 )  (x^2 /a^2 )−((2xycos α)/(ab))+(y^2 /b^2 ) =sin^2 α .
letθ+=αcos(θ+)=cosαcosθcossinθsin=cosα(xa)(yb)a2x2a.b2y2b=cosα(xyabcosα)2=(1x2a2)(1y2b2)2xycosαab+cos2α=1x2a2y2b2x2a22xycosαab+y2b2=sin2α.
Commented by rish@bh last updated on 03/Mar/17
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