cos(θ)=,  < θ < .
's
I changed your x's to θ so that x could represent the values on the
x-axis rather than the angle.

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We draw the angle θ in the second quadrant.




And we draw a perpendicular to the x-axis from the end of the green line
forming a right triangle.


The bottom side of the triangle is x
The vertical side of the triangle is y, 
and the green line is r.

The cosine is  and we are given that  cos(θ)=
so we will take x as the numerator, considering it negative, since it
goes to the left.  So we will label it x=-4, and we will label r as
the denominator of , considering it positive, since it goes
up.  So we will label it as r=5



We calculate y from

r² = x²+y²
5² = (-4)²+y²
25 = 16+y²
 9 = y²
±3 = y, we take the positive value since y goes up.

So we label the y-value as 3



sin(2θ) = 2sin(θ )cos(θ ) 

Since sin(θ ) = , sin(θ ) =   

sin(2θ) = 2sin(θ )cos(θ ) = 2 = 

cos(2θ) = cos²(θ)-sin²(θ) = - = - =  

tan(2θ) =  

Since the tangent is , tan(θ) =  = 

tan(2θ) =  =  

Multiply top and bottom by 16

tan(2θ) =  =  =  = 

Edwin