show that 6(sinx+cosx)p4+12(sinx-cosx)p2+8(sinp6x+cosp6x)=26 where p4,p2,p6are use for power4,power2(square),power6

6(sinx+cosx)⁴+12(sinx-cosx)²+8(sin⁶x+cos⁶x)

= 6(sin²x+cos²x+2sinxcosx)²+12(sin²x+cos²x-2sinxcosx)+8(sin²x+cos²x)³ - 24sin²xcos²x(sin²x+cos²x)

= 6(1+2sinxcosx)²+12(1-2sinxcosx)+8 -24sin²xcos²x

= 6(1+4sin²xcos²x+4sinxcosx)+12(1-2sinxcosx)+8 -24sin²xcos²x

= 6+24sin²xcos²x+24sinxcosx+12-24sinxcosx+8 -24sin²xcos²x

= 26

 

Hence, proved.