Hi. um I really don't know how to do this but can you plz help me?

I'm assuming that this is a trig identity that you are to verify:

[1 + cos(x) + cos(2x)] / [sin(x) + sin(2x)]  =?=  cot(x)

Since cos(2x)  =  cos²(x) - sin²(x):

     [ 1 + cos(x) + cos(2x) ] / [ sin(x) + sin(2x) ]   --->   [ 1 + cos(x) + cos²(x) - sin²(x) ] / [ sin(x) + sin(2x) ]

Since  1 - sin²(x)  =  cos²(x) :

     [ 1 + cos(x) + cos²(x) - sin²(x) ] / [ sin(x) + sin(2x) ]   --->   [ cos(x) + cos²(x) + cos²(x) ] / [ sin(x) + sin(2x) ] 

     Simplify:   --->   [ cos(x) + 2cos²(x) ] / [ sin(x) + sin(2x) ]  

     Factor:     --->   [ cos(x)[ 1 + 2cos(x) ] / [ sin(x) + sin(2x) ]  

Since  sin(2x)  =  2sin(x)cos(x):

     [ cos(x)( 1 + 2cos(x) ) ] / [ sin(x) + sin(2x) ]    --->   [ cos(x)( 1 + 2cos(x) ) ] / [ sin(x) + 2sin(x)cos(x) ]

     Factor:   --->   [ cos(x)( 1 + 2cos(x) ) ] / [ sin(x)( 1 + 2cos(x) ) ]

Cancelling the  1 + 2cos(x)  terms:

     --->   [ cos (x) ] / [ sin(x) ]

Since  cot(x)  =  cos(x) / sin(x)

     --->   cot(x)

Verify the following identity:
(1+cos(x)+cos(2 x))/(sin(x)+sin(2 x)) = cot(x)

Write cotangent as cosine/sine:
(1+cos(x)+cos(2 x))/(sin(x)+sin(2 x)) = ^?(cos(x))/(sin(x))

Cross multiply:
sin(x) (1+cos(x)+cos(2 x)) = ^?cos(x) (sin(x)+sin(2 x))

cos(2 x) = 1-2 sin(x)^2:
sin(x) (1+cos(x)+1-2 sin(x)^2) = ^?cos(x) (sin(x)+sin(2 x))

1+cos(x)+1-2 sin(x)^2 = 2+cos(x)-2 sin(x)^2:
2+cos(x)-2 sin(x)^2 sin(x) = ^?cos(x) (sin(x)+sin(2 x))

(2+cos(x)-2 sin(x)^2) sin(x) = 2 sin(x)+cos(x) sin(x)-2 sin(x)^3:
2 sin(x)+cos(x) sin(x)-2 sin(x)^3 = ^?cos(x) (sin(x)+sin(2 x))

sin(2 x) = 2 sin(x) cos(x):
2 sin(x)+cos(x) sin(x)-2 sin(x)^3 = ^?cos(x) (sin(x)+2 cos(x) sin(x))

cos(x) (sin(x)+2 cos(x) sin(x)) = cos(x) sin(x)+2 cos(x)^2 sin(x):
2 sin(x)+cos(x) sin(x)-2 sin(x)^3 = ^?cos(x) sin(x)+2 cos(x)^2 sin(x)

cos(x)^2 = 1-sin(x)^2:
2 sin(x)+cos(x) sin(x)-2 sin(x)^3 = ^?cos(x) sin(x)+2 1-sin(x)^2 sin(x)

2 (1-sin(x)^2) sin(x) = 2 sin(x)-2 sin(x)^3:
2 sin(x)+cos(x) sin(x)-2 sin(x)^3 = ^?cos(x) sin(x)+2 sin(x)-2 sin(x)^3

The left hand side and right hand side are identical:
Answer: |(identity has been verified)

(1 + cos(x) + cos(2x)) / (sin(x) + sin(2x)) = cot(x) ??

\(\begin{array}{|rcll|} \hline \cot (x) = \frac{1+\cos (2x)}{\sin(2x)} \\ \hline \end{array}\)  see unit-circle: \(\begin{array}{|rcll|} \hline \cot (\frac{x}{2}) = \frac{1+\cos (x)}{\sin(x)} \\ \hline \end{array}\)

\(\begin{array}{|rcll|} \hline \frac { 1 + \cos(x) + \cos(2x)} { \sin(x) + \sin(2x) } &=& \cot(x) \\\\ \frac { 1 + \cos(2x) + \cos(x) } { \sin(x) + \sin(2x) } &=& \cot(x) \\\\ 1 + \cos(2x) + \cos(x) &=& \cot(x) \cdot [ \sin(x) + \sin(2x) ] \quad | \quad 1+\cos (2x) = \sin(2x)\cdot \cot(x)\\ \sin(2x)\cdot \cot(x) + \cos(x) &=& \cot(x) \cdot [ \sin(x) + \sin(2x) ] \\ \sin(2x)\cdot \cot(x) + \cos(x) &=& \cot(x) \cdot \sin(x) + \cot(x) \cdot \sin(2x) \\ \cos(x) &=& \cot(x) \cdot \sin(x) \\ \frac{\cos(x)}{\sin(x)} &=& \cot(x) \\ \hline \end{array}\)