How does $d \sqrt{1+\left(\frac{h_t+h_r}{d}\right)^2}$ become $d \left(1+\frac{1}{2}\left(\frac{h_t+h_r}{d}\right)^2\right)$?

Question

I am trying to understand the following step related to square root in equation (2).

$$\Delta = d \sqrt{1+\left(\frac{h_t+h_r}{d}\right)^2} \tag1$$

$$\Delta = d \left(1+\frac{1}{2}\left(\frac{h_t+h_r}{d}\right)^2\right) \tag2$$

My query is that, I am not getting how square root disappears in equation (1) and $\frac{1}{2}$ appears in equation (2).

Any help in this regard will be highly appreciated.

Answer 1

If $x$ is small then $$ \sqrt{1+x} \approx 1 + \frac{x}{2} $$ since $$ \left(1 + \frac{x}{2} \right)^2 = 1 + x + \frac{x^2}{4} $$ and $x^2 << x$.

I suspect that in the context where your equations appear the second expression for $\Delta$ is an approximation, not an equality.