(1+2sin(2x)+cos(2x)) / (2+sin(2x)-2cos(2x)) =cot(x)

Necesito verificar esta identidad (1+2sin(2x)+cos(2x))/(2+sin(2x)-2cos(2x)… =cot(x) ayuda por favor gracias!!!

sin(2x) = 2sin(x)cos(x)
cos(2x) = cos^2(x) - sin^2(x)
1 = sin^2(x) + cos^2(x)

Numerator
(1 + 2sin(2x) + cos(2x))
(sin^2(x) + cos^2(x)) + (2 * (2sin(x)cos(x))) + (cos^2(x) - sin^2(x))
2cos^2(x) + 2sin(x)cos(x)
2cos(x) * (cos(x) + 2sin(x))

Denominator
(2 + sin(2x) - 2cos(2x))
2(sin^2(x) + cos^2(x)) + 2sin(x)cos(x) - 2(cos^2(x) - sin^2(x))
2sin^2(x) + 2cos^2(x) + 2sin(x)cos(x) - 2cos^2(x) + 2sin^2(x)
4sin^2(x) + 2sin(x)cos(x)
2sin(x) * (2sin(x) + cos(x))
2sin(x) * (cos(x) + 2sin(x))

2cos(x) * (cos(x) + 2sin(x)) / 2sin(x) * (cos(x) + 2sin(x))

Which reduces to

cos(x) / sin(x) = cot(x)

De nada.