Find cos2A in terms of sin B
To find cos2A in terms of sin B, we will first use the trigonometric identity:

cos2A = 1 - 2sin^2A

Given:
tan A = 1/2
tan B = 1/3

Finding sin A and cos A:
Since tan A = 1/2, we can let A be an angle in a right triangle where the opposite side is 1 and the adjacent side is 2. Therefore, sin A = 1/sqrt(1^2 + 2^2) = 1/sqrt(5) and cos A = 2/sqrt(5) = 2sqrt(5)/5.

Using the Pythagorean identity:
sin^2A + cos^2A = 1
(1/5) + (4/5) = 1
5/5 = 1

Substitute sin A and cos A into the cos2A formula:
cos2A = 1 - 2(sinA)^2
cos2A = 1 - 2(1/5)
cos2A = 1 - 2/5
cos2A = 3/5

Expressing sin B in terms of sin A:
We know that tan B = 1/3, so let B be an angle in a right triangle where the opposite side is 1 and the adjacent side is 3. Therefore, sin B = 1/sqrt(1^2 + 3^2) = 1/sqrt(10) = sqrt(10)/10.

Expressing cos2A in terms of sin B:
Since cos2A = 3/5 and sin B = sqrt(10)/10, we can conclude that cos2A = 3/5 = 6sqrt(10)/10 = 6sin B/5.
Therefore, cos2A can be expressed in terms of sin B as 6sin B/5.