Show that the ratio of the volumes of the two regular square pyramids is $\frac{\tan 40°}{\tan 80°}$.

The two regular square pyramids are shown below:

The formula for the volume of pyramid is the one third of the product of base area and height.

$\text{V}=\frac{1}{3}\text{B}h$

The length of the side of square base for both the pyramids is same. So, the square base area for both the pyramids is equal.

So, the ratio of volume of both the pyramids is the ratio of the heights of both the pyramids.

The height of the first pyramid is the product of base edge and the tangent of the angle given in pyramid.

$\begin{array}{c}{h}_1=\left(\text{base\hspace{0.33em}edge}\right)\tan 40°\\ =10\tan 40°\end{array}$

The height of the second pyramid is the product of base edge and the tangent of the angle given in pyramid.

$\begin{array}{c}{h}_2=\left(\text{base\hspace{0.33em}edge}\right)\tan 80°\\ =10\tan 80°\end{array}$

Now simplify the ratio of the volumes of both the pyramids.

$\begin{array}{c}\frac{{\text{V}}_1}{{\text{V}}_2}=\frac{h_1}{h_2}\\ =\frac{10\tan 40°}{10\tan 80°}\\ =\frac{\tan 40°}{\tan 80°}\end{array}$

Hence, the ratio of the volumes of two regular square pyramid is equal to $\frac{\tan 40°}{\tan 80°}$.