Which of the following expressions is equivalent to 2 - 4 cos²x + 4 cos⁴x? A)2 sin²x + 2 cos²x B)2 sin⁴x - 2 cos²x C)2 sin2x + 2 cos⁴x D)2 sin⁴x + 2 cos⁴x E)2 sin⁴x - 2 cos⁴x

Question

Quizplus · Accepted Answer

Use the identity cos(A+B) = cosAcosB - sinAsinB with A = 2x and B = -2x to simplify the expression. Then, we get:

2 - 4 cos(2x) + 4 cos(4x)
= 2 - 4(cos^2(x) - sin^2(x)) + 4(2cos^2(2x) - 1)
= 2sin^2(x) + 8cos^2(2x)
= 2(sin^2(x) + 4cos^2(2x))
= 2(sin^2(x) + 2cos^2(2x) + 2cos^2(2x))
= 2(sin(x)^2 + cos(2x)^2 + cos(2x)^2)
= 2(sin(x)^2 + 2cos(2x)^2)

Now, use the identity cos(2A) = 1 - 2sin^2(A) to simplify further:

2(sin(x)^2 + 2cos(2x)^2)
= 2(sin(x)^2 + 2(1 - cos(2x)^2))
= 2(sin(x)^2 + 2 - 2cos(2x)^2)
= 2(sin(x)^2 + 2 - (1 - 2sin^2(x)))
= 4sin^2(x) + 1

Finally, use the identity sin(2A) = 2sin(A)cos(A) with A = 2x to get:

4sin^2(x) + 1 = 2(sin(2x)^2 + 1)
= 2sin^2(2x) + 2cos^2(2x) + 2
= 2sin^2(2x) + 2 + 2cos^2(2x)
= 2(sin^2(2x) + cos^2(2x) + 1)
= 2(sin^2(2x) + cos^2(2x) + sin^2(x) + cos^2(x))
= 2(sin^2(x) + cos^2(x) + sin^2(2x) + cos^2(2x))
= 2(sin^2(x) + cos(x)^2)(1 + 2cos^2(x))
= 2(1 + 2cos^2(x))

Now, using the identity sin(A+B) = sinAcosB + cosAsinB with A = x and B = 2x, we get:

2(1 + 2cos^2(x))
= 2(1 + 2cos^2(x))(sin(x)^2 + cos(2x)^2)
= 2(1 + 2cos^2(x))(1 - sin(x)^2)
= 2(1 + 2cos^2(x))cos(x)^2
= 2cos(x)^2 + 4cos^4(x)

Therefore, the equivalent expression is: 2cos^2(x) + 4cos^4(x).

The only answer choice that matches this expression is D.

Which of the Following Expressions Is Equivalent to 2 -