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First of all, the question is cot(A+15)-tan(A-15) = 4cos2A / 1+2sin2A ( You have posted x instead of A)
Now taking L.H.S
cot(A+15) - tan(A-15)
cos(A+15) / sin(A+15) - sin(A-15) / cos(A - 15)
[cos(A+15)cos(A-15) - sin(A-15)sin(A+15)] / [ sin(A+15)cos(A-15)
cos2A / sin(A+15)cos(A-15)
2cos2A / 2sin(A+15)cos(A-15)
2cos2A / [sin2A +sin30]
2cos2A / [ sin2A + 1/2]
2cos2A / [(2sin2A+1)/2]
4cos2A / (2sin2A + 1)
= R.H.S
Hence proved....