Question

2tan^-1(tanx/2×tan(π/4-y/2))=tan^-1(sinx×siny/cosx+cosy)
You sure there are no typos? I suspect you want

(sinx+siny)/(cosx+cosy)

Anyway, you can get things a bit simpler by taking the tan of both sides, since

tan(arctan(x)) = x

also,

tan(x/2) = (1-cosx)/sinx

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