The values of x that are solutions to the equation cos^(2)x=sin2x in the interval [0, pi] are

a. arctan(1/2) only
b. arctan(1/2) and pi
c. arctan(1/2) and 0
d. arctan(1/2) and (pi/2)
e. arctan(1/2), o, and (pi/2)

cos^2 x - sin 2x = 0 , 0 ≤ x ≤ π

cos^2 x - 2sinxcosx= 0
cosx(cosx - 2sinx) = 0
cosx = 0
x = π/2
or
cosx = 2sinx
cosx/sinx = 2
sinx/cosx = 1/2
tanx = 1/2 , tan is positive only in I for our domain
x = arctan(1/2)

so it looks like D

To find the values of x that are solutions to the equation cos^2(x) = sin(2x), we can use trigonometric identities and solve step by step.

Step 1: Start with the given equation cos^2(x) = sin(2x).

Step 2: Rewrite sin(2x) using the double angle identity sin(2x) = 2sin(x)cos(x). The equation becomes: cos^2(x) = 2sin(x)cos(x).

Step 3: Rearrange the equation to have zero on one side: cos^2(x) - 2sin(x)cos(x) = 0.

Step 4: Factor out a common term: cos(x)(cos(x) - 2sin(x)) = 0.

Step 5: Now we have two possible solutions: cos(x) = 0 or (cos(x) - 2sin(x)) = 0.

For the equation cos(x) = 0, we know that x = (pi/2) or x = (3pi/2) are solutions within the interval [0, pi].

For the equation cos(x) - 2sin(x) = 0, let's solve it separately.

Step 6: Divide the equation by cos(x): 1 - 2(sin(x)/cos(x)) = 0. Simplify it to 1 - 2tan(x) = 0.

Step 7: Rearrange the equation to have zero on one side: 2tan(x) - 1 = 0.

Step 8: Solve for tan(x): tan(x) = 1/2.

Step 9: To find the values of x for which tan(x) = 1/2 in the interval [0, pi], we can use the inverse tangent function (arctan).

Step 10: Take the arctan of both sides: x = arctan(1/2).

Therefore, the values of x that are solutions to the equation cos^2(x) = sin(2x) in the interval [0, pi] are x = (pi/2), x = (3pi/2), and x = arctan(1/2).

Looking at the given answer choices, the correct answer is e. arctan(1/2), 0, and (pi/2).