How to calculate the real Ground Speed from True Air Speed?
First calculate horizontal component of airspeed, then add the wind:

$$v_{GS} = cos(\theta) * v_{TAS} + v_{wind}$$ with $\theta$ being the angle between the horizon and the path of the aircraft in the vertical plane.

Or, if you are unfamiliar with trigonometry (using Pythagora's theorem):

$$ v_{GS} = \sqrt{v_{TAS}^2-v_{verticalSpeed}^2} + v_{wind}$$

Both formulas assume the same units being used for all speeds ($v_{TAS}$, $v_{verticalSpeed}$, $v_{wind}$), and only take horizontal wind into consideration. $v_{wind}$ is only considering the headwind/tailwind component.