I read the left-hand side as: (1+sec(x))/((tan(x))^2+tan(x)sin(x))=cot(x)(1+sec(x))/(tan(x)+sin(x)).
Numerator=cos(x)/sin(x)+1/sin(x)=(cos(x)+1)/sin(x);
denominator=sin(x)(1/cos(x)+1). Whole fraction is: cos(x)(cos(x)+1)/((sin(x))^2(1+cos(x)).
The common factor (1+cos(x)) cancels out to leave:
cos(x)/(sin(x))^2=cos(x)/sin(x) * 1/sin(x)=cot(x)cosec(x) QED.