Question #f61d0

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Answer 1 · 2017-05-20T13:24:28

#LHS=(sin^3x-cos^3x)/(sinx+cosx)#

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

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

#=((sin^2x+cos^2x-2sinxcosx)(1+sinxcosx))/((sinx-cosx)(sinx+cosx))#

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

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

#=(1/sin^2x-(2sinxcosx)/sin^2x+(sinxcosx)/sin^2x-2cos^2x)/(1-cot^2x)#

#=(1/sin^2x-2cotx+cotx-2cos^2x)/(1-cot^2x)#

#=(csc^2x-cotx-2cos^2x)/(1-cot^2x)=RHS#

Proved