Mathematics

Problem

Let

f(x)=\csc x

for all

x

in the domain of

\csc

. (a) Express

f(x)

as the reciprocal of a single elementary trigonometric function. (b) Differentiate

f(x)

with respect to

x

and write the derivative using only

\sin x

and

\cos x

. (c) Demonstrate algebraically that the derivative found in part (b) simplifies to

-\csc x\cot x

. Provide a clear sequence of algebraic steps leading to the final form.