It cannot be proven because it is not true. You can prove that by choosing x and y that satisfies sin2x=3sin2y. For example, take x = 15 degrees and y = 9.5941 degrees.
For those choices, sin 2x = 3sin 2y = 0.500
For those choices of x and y
2 tan(x-y) = 0.1893 and tan (x+y) = 0.4577
By being able to find ANY x,y combination that satisfies sin2x=3sin2y, and for which your second equation is not true, then your contention is disproved.
Either someone is playing a practical joke on us, or on you, or you have copied the problem incorrectly
I made a mistake in my example. Choose the angles x = 15 degrees and y = 4.79703 degrees. Then the condition
sin 2x = 3 sin 2y is satisfied.
2tan(x-y) = 0.35996
tan (x+y) = 0.35996
Your identity is almost certainly correct and I must apologize for my error in the previous answer.
I tried to prove the second equation for any x and y that satisfy your first equation, but was not able to.
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