Simplify (sin x + cost)² + (sin x- cos x)²?

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Answer 1 · 2018-02-19T06:19:37

The result is #2#.

#color(white)=(sinx+cosx)^2+(sinx-cosx)^2#

#=sin^2x+2sinxcosx+cos^2x+(sinx-cosx)^2#

#=stackrel1 overbrace(sin^2x+cos^2x)+2sinxcosx+(sinx-cosx)^2#

#=1+2sinxcosx+(sinx-cosx)^2#

#=1+2sinxcosx+sin^2x-2sinxcosx+cos^2x#

#=1+2sinxcosx+stackrel1 overbrace(sin^2x+cos^2x)-2sinxcosx#

#=1+2sinxcosx+1-2sinxcosx#

#=2+color(red)cancel(color(black)(2sinxcosx-2sinxcosx))#

#=2#

Answer 2 · 2018-02-19T06:19:55

2

I suppose the sum is #(sinx + cosx)^2 + (sine - cosx)^2#

#(sinx + cosx)^2 + (sine - cosx)^2 = sin^2x + cos^2x + 2 sinx cosx + sin^2x + cos^2x - 2 sinx cosx#

#=> 1 + cancel(2 sinx cosx )+ 1 - cancel(2 sinx cosx)# as #(sin^2x + cos^2x = 1)#

#=> 2#