Please help me with this question.

Question

Differentiate y = csc(x) − cot(x).

Answer 1

recall d/dx csc x = -csc x cot x and d/dx cot x = -csc² x.

 

So dy/dx = -csc x cot x + csc²x = (csc x - cot x) csc x

 

Or you can break it into sines and cosines

 

dy/dx = d/dx [(1/sin x) - (cos x)/sin x]

 

= d/dx (1 - cos x)/sin x

 

= [(sin x)(sin x) - (1 - cos x)(cos x)]/sin² x

 

= (sin² x + cos² x - cos x)/sin² x

 

= (1 - cos x)/sin² x

 

= (csc x - cot x) csc x