While the standard method for proving a trigonometric identity is to manipulate only one side, it's also a valid strategy to simplify both sides of the equation separately until they meet at an identical expression.
Because the natural range of basic sine and cosine waves is strictly bounded between -1 and 1, equations setting them equal to numbers outside this window have no real solutions.
Unlike an ellipse, where the major axis is always strictly longer than the minor axis, a hyperbola's transverse axis can be shorter than, longer than, or equal in length to its conjugate axis.
Despite their similar-looking notation, the inverse sine function used to determine an angle is fundamentally different from the reciprocal of sine, which is known as cosecant.
