Working with the right side:
cot(x) + 2 tan(x) + tan³(x) = cos(x)/sin(x) + 2 sin(x)/cos(x) + sin³(x)/cos³(x)
… = (cos⁴(x) + 2 sin²(x) cos²(x) + sin⁴(x)) / (sin(x) cos³(x))
Factorize the numerator as a sum of squares:
a⁴ + 2 a² b² + b⁴ = (a² + b²)²
… = (cos²(x) + sin²(x))² / (sin(x) cos³(x))
Recall that
cos²(x) + sin²(x) = 1
… = 1 / (sin(x) cos³(x))
… = 1 / (sin(x) cos³(x)) • cos(x)/cos(x)
… = cos(x) / (sin(x) cos⁴(x))
… = cot(x) sec⁴(x)
