If Sin x - cos x = 1/3 , what is sin 3x + cos 3x?

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Answer 1 · 2016-08-23T06:04:19

#-25/27#

#sinx-cosx=1/3#

Squaring both sides

#=>(sinx-cosx)^2=1/9#

#=>sin^2x+cos^2x-2sinxcosx=1/9#

#=>1-2sinxcosx=1/9#

#=>2sinxcosx=1-1/9=8/9#

#=>sinxcosx=4/9#

Now
#sin3x+cos3x#

#=3sinx-4sin^3x+4cos^3x-3cosx#

#=3(sinx-cosx)-4(sin^3x-cos^3x)#

#=3xx1/3-4(sinx-cosx)(sin^2x+cos^2x+sinxcosx)#

#=1-4xx1/3(1+4/9)#
#=1-4/3xx13/9#
#=1-52/27=-25/27#