Problem. Show that a triangle is a right triangle if and only if
Solution. (By removeablesingularity)
Suppose that is a a right triangle. Then . Check that
and hence the equation will be satisfied.
Now, suppose that the equation is satisfied. If the left-hand-side (LHS) is squared, we obtain
and if the right-hand-side (RHS) is squared, we obtain
Recall the trigonometric identities:
- (for instance, )
The equation above after squared then can rewritten as
Let us write . Then,
which factorizes to
Hence, or or which consecutively implies , , or . Hence, if the equation is satisfied, triangle has to be a right triangle.