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Prove that this are equal: cos^2x+cos^2y=1+cos(x+y)*cos(x-y) ?

Answer 1

Anna Andrews

See below.

Start by expanding the right hand side using #cos(A + B) = cosAcosB - sinAsinB# and #cos(A - B) = cosAcosB + sinAsinB#.

#= 1 + (cosxcosy - sinxsiny)(cosxcosy + sinxsiny)#

#=1 + cos^2xcos^2y - sin^2xsin^2y#

#= 1 + cos^2xcos^2y - (1- cos^2x)(1 - cos^2y)#

#= 1 + cos^2xcos^2y - (1 - cos^2x - cos^2y + cos^2xcos^2y)#

#=1 + cos^2xcos^2y - 1 + cos^2x + cos^2y - cos^2ycos^2x#

#= cos^2x +cos^2y#

Now we see that #LHS = RHS# so the identity has been proved.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.