Have any of you seen this theorem of relationship of the angles between two slant faces of a pyramid and the angles between slant vectors, provided that two faces of corresponding to $\phi$ and $\eta$ are perpendicular? I attach the picture The result of theorem is $\cos(\theta)=\cos(\phi)\cdot cos(\eta)$.
Applying Pythagoras theorem to the base, we have $L^2 = a^2 \sin^2 φ + c^2 \sin^2 η$
Applying cosine law to the slant face, we have $L^2 = a^2 + c^2 – 2ac \cos θ$
∴ $a^2 \sin^2 φ + c^2 \sin ^2 η = a^2 + c^2 – 2ac \cos θ$
$2ac \cos θ = a^2 – a^2 \sin^2 φ + c^2 – c^2 \sin^2 η$
$2ac \cos θ = a^2 \cos^2 φ + c^2 \cos^2 η$
$2 ac \cos θ = (a \cos φ) ^2 + (c \cos η) ^2$
$2 ac \cos θ = 2 b^2$
$\cosθ = \frac {b}{a} \frac {b}{c}$
∴ $\cosθ = (\cosφ)( \cos η)$
Very good. I was looking for a proof from the Pythagoras theorem. Any way I proved in a simple way by means of inner products. You can look this. Later I realized it is a special case of law of cosines...
