Chain Rule:
let u=2x and y=tan(u)
du/dx = 2 and dy/du = sec^2(u)
dy/dx = du/dx x dy/du
multiply them together and replace u=2x into the equation..
therefore dy/dx = 2(sec^2(2x))
hope that helps.
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tan(2x) = 2tanx/(1 - tan2x) So tan(2x) +1 = 2tanx/(1 - tan2x) + 1 = 2tanx/(1 - tan2x) + (1 - tan2x)/(1 - tan2x) = (-tan2x + 2tanx + 1)/(1 - tan2x)
It depends on the domain of x.
(1 - tan2x)/(1 + tan2x) = (1 - sin2x/cos2x)/(1 + sin2x/cos2x) = (cos2x - sin2x)/(cos2x + sin2x) = (cos2x - sin2x)/1 = (cos2x - sin2x) = cos(2x)
I assume you mean (tanx+1)^2 In which case, (tanx+1)^2=tan2x+2tanx+1
The period of the tangent function is PI. The period of y= tan(2x) is PI over the coefficient of x = PI/2