$h$ = distance from ground to eye of a person, who measured the angle to the top of the building with a protractor.
$θ$ = from the eye to the top of the building.
$x$ = distance of the person from the building.
$y$ = the height of the building
The equation to find the height of the building is given by: $y = h + x ⋅ \tan(θ)$
The following introduces the error values and solves for $Δy$:
$$\tan(θ + Δθ) = \frac{(y + Δy) - (h + Δh)}{x + Δx}$$
$$\tan(θ + Δθ) = \frac{Δy - Δh + (y - h)}{x + Δx}$$
$$\tan(θ + Δθ) = \frac{Δy - Δh + x ⋅ tan(θ)}{x + Δx}$$ --> Uses the fact that $\tan(θ) = \frac{(y - h)}{x}$
$$Δy = Δh + (x + Δx) ⋅ \tan(θ + Δθ) - x ⋅ \tan(θ)$$
Now, the course I'm taking goes on to say "to get a sense of the order of the error, let's find the linear approximation of $\tan(θ + Δθ)$" and then plugs it in the same place so that:
$Δy \approx Δh + (x + Δx)(\tan(θ) + \sec^2(θ)Δθ) - x ⋅ \tan(θ)$
My question is, why is this substitution done? How does it affect the formula? Why does it help solve the for the error? Why find the linear approximation for this but not any other term in the formula? The course I'm taking does not adequately explain why this needs to be done.
Summarizing the discussion in the comments:
The point here is to achieve a linear approximation. That is we want an expression of the form $$f(\theta +\Delta\theta)-f(\theta)\approx C(\theta) \Delta \theta$$
Where $C(\theta)$ may depend on $\theta$, or any of the other parameters of the problem, but not on $\Delta \theta$.
The idea behind this is that we wish to avoid having to evaluate transcendental functions, like the trig functions, anywhere except at special values. The approximation gives us am easily computable way of estimating the value of those functions near special values.