By definition, tan^(-1)(4/3) = (theta) means that tan(theta) = 4/3. So draw a right triangle, label the one angle as "theta", and fill in the "opposite" and "adjacent" values. Use the Pythagorean Theorem to find the length of the remaining side. Then take the cosine.

How would you start the solution to 2XY + X = 0? Do the same factorization here.

Convert the left-hand side to sines and cosines, using the double-angle identity they gave you. Convert the right-hand side to sines and cosines using the definition of tangent. See where that leads....

How so? (You might have been "nearly there", but until we can see your work, we just can't know where you're needing the help.)