Apparently, I can multiply this by a fraction equal to one that contains some trig functions.
I can't seem to find this "trick" multiplier simplify this. I've tried cosx/cosx , cscx/cscx, cotx/cotx, etc.
Here, you have 1+sin(x)
Try to find identities with 1 and sin(x). The first that comes in mind is cosē(x)+sinē(x)=1 ---> 1-sinē(x)=cosē(x)=(1-sin(x))(1+sin(x))
When you're dealing with cot, or tan, it can be interesting to first multiply by cos or sin, regarding the situation.
When you're dealing with a sum of cos or a sum of sin, recall the identities of addition sin(a+b), cos(a+b)... cos(a)+cos(b)=.... sin(a)+sin(b)=....
But it's true, like TPH says, a lot comes with experience.