cos²(x-π/6)-sin²x
= [cos(x-π/6)-sinx] [cos(x-π/6)+sinx]
=[cosxcosπ/6+sinxsinπ/6-sinx][cosxcosπ/6+sinxsinπ/6+sinx]
=[cosxcosπ/6+1/2sinx-sinx][cosxcosπ/6+1/2sinx+sinx]
=[cosxcosπ/6-1/2sinx][√3/2cosx+3/2sinx]
=√3[cosxcosπ/6-sinxsinπ/6][1/2cosx+√3/2sinx]
=√3[cosxcosπ/6-sinxsinπ/6][sinπ/6cosx+cosπ/6sinx]
=√3cos(x+π/6)sin(π/6+x)
=√3cos(x+π/6)sin(π/6+x)
=√3/2sin(π/3+2x)