Answer
See explanations.
Work Step by Step
a. $\frac{d}{dx}(sec(x))=\frac{d}{dx}(\frac{1}{cos(x)})=\frac{cos(x)(0)-(-sin(x))}{cos^2(x)}=\frac{sin(x)}{cos^2(x)}=sec(x)tan(x)$
b. $\frac{d}{dx}(csc(x))=\frac{d}{dx}\frac{1}{sin(x)}=\frac{siin(x)(0)-cos(x)}{sin^2(x)}=\frac{-cos(x)}{sin^2(x)}=-csc(x)cot(x)$
c. $\frac{d}{dx}(cot(x))=\frac{d}{dx}(\frac{cos(x)}{sin(x)})=\frac{sin(x)(-sin(x))-cos(x)(cos(x))}{sin^2(x)}=-\frac{sin^2(x)+cos^2(x)}{sin^2(x)}=-\frac{1}{sin^2(x)}=-csc^2x$
