Please need trig help

[samantha ryan]

1. complete the identity: x=theta
sin (2x)tan x + cos (2x)=?

2. Establish the identity: x=theta
csc (2x) - cot (2x)= tan x

 

[soroban]

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: .$\displaystyle \frac{1}{\sin(2x)}\,-\,\frac{\cos(2x)}{\sin(2x)}\;=\;\frac{1\,-\,\cos(2x)}{\sin(2x)}\;=\;\frac{2\cdot\sin^2(x)}{2\cdot\sin(x)\cdot\cos(x)}\;=\;\frac{\sin(x)}{\cos(x)}\;=\;\tan(x)$

 

[samantha ryan]

Hi Soroban,

THANK YOU! THANK YOU! THANK YOU! now i understand it better! thank you again soooooooo much!