Hello, samantha!
You're expected to know several identities . . . among them are:
. . . \(\displaystyle \sin(2x)\:=\:2\cdot\sin(x)\cdot\cos(x)\)
. . . \(\displaystyle \cos(2x)\:=\:1\,-\,2\sin^2(x)\;\;\Rightarrow\;\;1\,-\,\cos(2x)\:=\:2\cdot\sin^2(x)\)
1. Complete the identity: . \(\displaystyle \sin(2x)\tan(x)\,+\,\cos(2x)\:=\;?\)
We have:
.\(\displaystyle [2\cdot\sin(x)\cdot\cos(x)]\cdot\left[\frac{\sin(x)}{\cos(x)}\right]\,+\,[1\,-\,2\cdot\sin^2(x)]\)
. . . \(\displaystyle =\;2\cdot\sin^2(x)\,+\,1\,-\,2\cdot\sin^2(x)\;=\;1\)
2. Establish the identity: .\(\displaystyle \csc(2x)\,-\,\cot(2x)\:=\:\tan(x)\)
The left side is:
.\(\displaystyle \frac{1}{\sin(2x)}\,-\,\frac{\cos(2x)}{\sin(2x)}\;=\;\frac{1\,-\,\cos(2x)}{\sin(2x)}\;=\;\frac{2\cdot\sin^2(x)}{2\cdot\sin(x)\cdot\cos(x)}\;=\;\frac{\sin(x)}{\cos(x)}\;=\;\tan(x)\)
