1/sin(x)+1/cos(x)≡(sin(x)+cos(x))/(sin(x)cos(x))≡
2(sin(x)+cos(x))/sin(2x). “≡” means “identical to”.
Now sin(x)+cos(x)≡(cos²(x)-sin²(x))/(cos(x)-sin(x))≡
cos(2x)/(cos(x)-sin(x)).
But cos(2x)/sin(2x)≡cot(2x).
So 2(sin(x)+cos(x))/sin(2x)≡
2cot(2x)/(cos(x)-sin(x)).
So 1/sin(x)+1/cos(x)≡2cot(2x)/(cos(x)-sin(x)).
This is one trig identity but there may be others.