Ftheta 3 cos 2theta 2sin 2theta2 3 as a sum of powers of s

F(theta) = 3 cos 2theta + 2(sin 2theta)^2 +3 as a sum of powers of sin theta. Answer choices f(theta) = 6 - 2 sin^2 theta + 8 sin^4 theta f(theta) = 6 - 8 sin^2 theta + 2 sin^4 theta f(theta) = 6 - 2 sin^2 theta - 8 sin^4 theta f(theta) = 6 + 2 sin^2 theta - 8 sin^4 theta f(theta) = 6 + 2 sin^2 theta + 8 sin^4 theta

Solution

Given

f() = 3 cos(2) + 2 ( sin2 )^2 + 3

we know cos(2) = 1 - 2 sin^2()

sin(2) = 2 sin() cos()

  (sin(2))^2 = 4 sin^2() cos^()

= 3 (1 - 2 sin^2() )+ 2 ( sin2 )^2 + 3

= 3 - 6 sin^2() + 2 ( 4 sin^2() cos^() )+ 3

= 6 -  6 sin^2() + 8 sin^2() cos^()

we know cos^() = 1- sin^2()

==> 6 -  6 sin^2() + 8 sin^2() (1- sin^2())

==> 6 - 6 sin^2() + 8 sin^2() - 8 sin^4()

==> 6 + 2 sin^2() -  8 sin^4()