cos(2alpha)cos(alpha) + cos(alpha) + sin(alpha)sin(2alpha)
Expanding cos(2alpha) using the double angle identity:
(2cos(alpha)^2 - 1)cos(alpha) + cos(alpha) + sin(alpha)2sin(alpha)*cos(alpha)
Apply trigonometric identities:
2cos(alpha)^3 - cos(alpha) + cos(alpha) + 2sin(alpha)^2*cos(alpha)
Simplify further:
2cos(alpha)^3 + cos(alpha) + 2sin(alpha)^2*cos(alpha)
Can be simplified to:
cos(alpha)(2cos(alpha)^2 + sin(alpha)^2 + 1)
cos(2alpha)cos(alpha) + cos(alpha) + sin(alpha)sin(2alpha)
Expanding cos(2alpha) using the double angle identity:
(2cos(alpha)^2 - 1)cos(alpha) + cos(alpha) + sin(alpha)2sin(alpha)*cos(alpha)
Apply trigonometric identities:
2cos(alpha)^3 - cos(alpha) + cos(alpha) + 2sin(alpha)^2*cos(alpha)
Simplify further:
2cos(alpha)^3 + cos(alpha) + 2sin(alpha)^2*cos(alpha)
Can be simplified to:
cos(alpha)(2cos(alpha)^2 + sin(alpha)^2 + 1)