Start with the identity cos^2(theta) + sin^2(theta) = 1
Divide both sides by cos^2(theta):
cos^2(theta)/cos^2(theta) + sin^2(theta)/cos^2(theta) = 1/cos^2(theta)
Use the identity sin(theta)/cos(theta) = tan(theta) and cos^2(theta)/cos^2(theta) = 1:
1 + tan^2(theta) = 1/cos^2(theta)
Solve for cos(theta):
cos^2(theta) = 1/(1 + tan^2(theta))
cos(theta) = +-sqrt(1/(1 + tan^2(theta)))
The domain restriction pi < theta < (3pi)/2 tells us that theta is in the third quadrant (where the cosine function is negative), therefore, we change the +- to - only:
cos(theta) = -sqrt(1/(1 + tan^2(theta)))
Substitute (2/3)^2 for tan^2(theta)
cos(theta) = -sqrt(1/(1 + (2/3)^2))
cos(theta) = -sqrt(1/(1 + 4/9))
cos(theta) = -sqrt(1/(13/9))
cos(theta) = -sqrt(9/13)
cos(theta) = -(3sqrt(13))/13
