Question

Verify the identity
(1+csc 3B)/(sec 3B) - cot 3B = cos 3B

Answer 1

The 3B is just a red herring
\[\frac{1+csc(3B)}{sec(3B)}-cot(3B) = cos(3B)(1+\frac{1}{sin(3B)})-\frac{cos(3B)}{sin(3B)}\]
\[=cos(3B)(\frac{sin(3B)}{sin(3B)} + \frac{1}{sin(3B)})-\frac{cos(3B)}{sin(3B)}\]
\[=\frac{cos(3B)sin(3B) + cos(3B)}{sin(3B)} - \frac{cos(3B)}{sin(3B)}\]
\[=\frac{cos(3B)sin(3B)}{sin(3B)} = cos(3B)\]

Answer 2

The operative identities:
\[csc(x) = 1/sin(x)::sec(x) = 1/cos(x):: cot(x)=cos(x)/sin(x)\]