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Question:

Find the value of the following: tan(1/2)[sin(2x/(1+x²)) + cos((1-y²)/(1+y²))], |x| < 1, y < 0 and xy < 1

Solution:

tan(1/2)[sin(2x/(1+x²)) + cos((1-y²)/(1+y²))] = tan(1/2)[2tan⁻¹x + 1/2cos(2tan⁻¹y)] = tan(tan⁻¹x + tan⁻¹y) = (x+y)/(1-xy)