tan (x+v)

Please note that,

since sin x= 2/3, then cos x= sqrt(1- 4/9) = sqrt(5)/3 so cos x = sqrt(5)/3, and tan x= (2/3) / (sqrt(5)/3)

then tan x= 2/sqrt(5)

furthermore we have:

cos v= -1/4 then sin v= sqrt(1- 1/16) = sqrt(15)/4

then

tan v= (sqrt(15)/4 / (-1/4), or:

tan v= -sqrt(15)

Next I apply the formula of addition for the tangent function and I get:

tan (x+v) = (tan x+ tan v)/(1- tan x * tan v)= [2/sqrt(5) -sqrt(15)]/[1-(2/sqrt(5) * -sqrt(15))]=

= (2-5 sqrt(3)) / (sqrt(5) * (1+ 2 sqrt(3)),

finally, we have:

tan (x+v) = [2-5*sqrt(3)] / [sqrt(5) * (1+ 2 sqrt(3))]

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