LHS=sin^6x-cos^6xLHS=sin6x−cos6x
=(sin^2x)^3-(cos^2x)^3=(sin2x)3−(cos2x)3
=[sin^2x-cos^2x]*[(sin^2x)^2+sin^2x*cos^2x+(cos^2x)^2]=[sin2x−cos2x]⋅[(sin2x)2+sin2x⋅cos2x+(cos2x)2]
=[sin^2x-(1-sin^2x)][sin^4x+sin^2x*cos^2x+cos^4x]=[sin2x−(1−sin2x)][sin4x+sin2x⋅cos2x+cos4x]
=[sin^2x-1+sin^2x][sin^4x+cos^2x(sin^2x+cos^2x)]=[sin2x−1+sin2x][sin4x+cos2x(sin2x+cos2x)]
=[2sin^2x-1][sin^4x+cos^2x]=RHS=[2sin2x−1][sin4x+cos2x]=RHS