Verify (cosx – cosy / sinx + siny) + (sinx – siny / cosx + cosy) = 0?

4 Answers

• (cosx - cosy / sinx + siny) + (sinx - siny / cosx + cosy) = 0 =====> LCD

  [ (cosx - cosy) * (cosx + cosy) / (sinx + siny) * (cosx + cosy) + [ (sinx - siny)*(sinx + siny) / (cosx + cosy) * (sinx + siny) ] = 0

  [ ( cos^2(x) + cos(x)cos(y) - cos(y)cos(x) - cos^2(y) ) / (sinx + siny) * (cosx + cosy) + [ (sin^2(x) + sin(x)sin(y) - sin(y)sin(x) - sin^2(y)) / (cosx + cosy) * (sinx + siny) ]

  [ ( cos^2(x) + cos(x)cos(y) - cos(y)cos(x) - cos^2(y) ) + ( sin^2(x) + sin(x)sin(y) - sin(y)sin(x) - sin^2(y) ) ] / [ (cosx + cosy) * (sinx + siny) ]

  [ cos^2(x) - cos^2(y) + sin^2(x) - sin^2(y) ] / [ (cosx + cosy) * (sinx + siny) ]

  [ cos^2(x) + sin^2(x) - cos^2(y) - sin^2(y) ] / [ (cosx + cosy) * (sinx + siny) ]

  knowing the identities;

  cos^2(x) + sin^2(x) = 1

  [ 1 - cos^2(y) - sin^2(y) ] / [ (cosx + cosy) * (sinx + siny) ]

  [ 1 - ( cos^2(y) + sin^2(y) ) ] / [ (cosx + cosy) * (sinx + siny) ]

  [ 1 - ( 1 ) ] / [ (cosx + cosy) * (sinx + siny) ]

  [ 1 - 1 ] / [ (cosx + cosy) * (sinx + siny) ]

  [ 0 ] / [ (cosx + cosy) * (sinx + siny) ]

  0

• Sinx Siny

  Source(s): https://shrinke.im/a8nZA

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  [(cosx-cosy)/(sinx+siny)] + [(sinx-siny)/(cosx+cosy)] = [cos^2x - cos^2 y + sin^2x - sin^2y]/[(sinx + siny)(cosx + cosy)] = [1-1]/[(sinx + siny)(cosx + cosy)] = 0 -------- Attn: Since [(cosx-cosy)/(sinx+siny)] + [(sinx-siny)/(cosx+cosy)] is given, that means it is well defined and (sinx+siny)(cosx+cosy) not = 0

• (cosx - cosy / sinx + siny) + (sinx - siny / cosx + cosy)

  = [(cos^2x - cos^2y) + (sin^2x - sin^2y)] / [(sinx + siny)(cosx + cosy)]

  = [cos^2x+sin^2x - (cos^2y+sin^2y)] / [(sinx + siny)(cosx + cosy)]

  = (1 - 1)/ [(sinx + siny)(cosx + cosy)]

  = 0