We have: (csc(- x)) / (cot(- x))csc(−x)cot(−x)
Let's apply the fact that csc(x)csc(x) and cot(x)cot(x) are odd functions:
= (- csc(x)) / (- cot(x))=−csc(x)−cot(x)
= (csc(x)) / (cot(x))=csc(x)cot(x)
Then, let's apply two standard trigonometric identities; csc(x) = (1) / (sin(x))csc(x)=1sin(x) and cot(x) = (cos(x)) / (sin(x))cot(x)=cos(x)sin(x):
= ((1) / (sin(x))) / ((cos(x)) / (sin(x)))=1sin(x)cos(x)sin(x)
= (1) / (sin(x)) cdot (sin(x)) / (cos(x))=1sin(x)⋅sin(x)cos(x)
= (1) / (cos(x))=1cos(x)
Finally, let's apply another standard trigonometric identity; sec(x) = (1) / (cos(x))sec(x)=1cos(x):
= sec(x)=sec(x)
