Q:

Elementary trigonometry and the right-angled triangle.

We have a right-angled triangle here. Angle ABD is $90^\circ$ while $\angle BCD$ is $\theta$.

Statement: Hypotenuse doesn't change when changing the observation side. Now when $\angle BCD$ is $\theta$, so  $BC$ the base and $AD$ the perpendicular such that the area of $\triangle ABC$ can be written as $S = \frac {BC \cdot BD}{2}$.

Is the statement true?

A:

There are two separate statements here!
"Hypotenuse doesn't change when changing the observation side".  I am not familiar with the term "observation side" but I presume that has to do with which side is the "near side" and which is the "opposite side".  The hypotenuse is the unique side opposite the right angle so this statement is true.
"The area of a right triangle can be written $\frac{BC\cdot BD}{2}$."  Yes, that is also true.  The formula for the area of a triangle is "1/2 base times height" where the "base" is the length of one side of the triangle and the "height" is the distance from the third vertex to that side, measured along a line perpendicular to that side.  But in a right triangle that "perpendicular" is the other side.
Another way to see this is to imagine the right triangle "duplicated" on the other side of the hypotenuse.  That is, imagine a new point, D, such that AD is parallel to BC and the same length as BC.  Then CD is also parallel to AB and the same length as AB.  "ABCD" is a rectangle with side lengths AB and BC.  It's area is $AB\cdot BC$.  The original triangle, ABC, is half of that rectangle so its area is $\frac{AB\cdot BC}{2}$.

