Recall that,
cos3A=4cos^3A-3cosA, &, sin3A=3sinA-4sin^3A.cos3A=4cos3A−3cosA,&,sin3A=3sinA−4sin3A.
Sub.ing these, we have,
:." The L.H.S.="cos^3A(4cos^3A-3cosA)+sin^3A(3sinA-4sin^3A),
=4cos^6A-3cos^4A+3sin^4A-4sin^6A,
=4(cos^6A-sin^6A)-3(cos^4A-sin^4A),
=4{(cos^2A)^3-(sin^2A)^3}-3{(cos^2A)^2-(sin^2A)^2},
=4{cos^2A-sin^2A){(cos^2A)^2+cos^2Asin^2A+(sin^2A)^2}
-3(cos^2A-sin^2A)(cos^2A+sin^2A),
=(cos^2A-sin^2A)[4{(cos^2A)^2+cos^2Asin^2A+(sin^2A)^2}-3*1],
=(cos^2A-sin^2A)[4cos^4A+4cos^2Asin^2A+4sin^4A-3(cos^2A+sin^2A)^2],
=(cos^2A-sin^2A)[4cos^4A+4cos^2Asin^2A+4sin^4A-3(cos^4A+2cos^2Asin^2A+sin^4a)],
=(cos^2A-sin^2A)[cos^4A-2cos^2Asin^2A+sin^4A],
=(cos^2A-sin^2A)(cos^2A-sin^2A)^2,
=(cos^2A-sin^2A)^3,
=(cos2A)^3,
=cos^3 2A,
"=The R.H.S."
Enjoy Maths.!