((cot x + 1)(cot x + 1) - csc 2 x))/cosx=?

Question

I'm not sure what to do to complete the identity of the problem. i also want to know how you know what identity to use like Reciprocal, Pythagorean, and Quotient.

Answer 1

(cot^2 + 2*cot + 1 - csc 2x )/cos

[(1/tan)^2 + 2 ( 1/tan) + 1 - 1/sin(2x) ]* 1/cos

[(1 + 2*tan + tan^2)/tan^2 - 1/(2sin*cos) ]* 1/cos

[ ( 1 + tan)^2/tan^2 - 1/(2*sin*cos)]* 1/cos

[ (1 + tan )^2 / tan^2 - (2*sin) ]

(1 + tan)^2 * (cos/sin)^2 - 2*sin

(1 + sin/cos)^2 * (cos/sin)^2 - 2*sin

(1 + 2*sin/cos + sin^2/cos^2) * (cos^2/sin^2) - 2*sin

( cos^2/sin^2 + 2 * cos/sin + 1) - 2*sin

( cos^2 + 2*cos*sin + sin^2)/sin^2 - 2 sin

(cos + sin)^2/sin^2 - 2*sin^3/sin^2

[(cos + sin)^2 - 2*sin^3]/sin^2