Problem 92 - Precalculus Essentials

Question

Graph each side of the equation in the same viewing rectangle. If the graphs appear to coincide, verify that the equation is an identity. If the graphs do not appear to coincide, this indicates that the equation is not an identity. In these exercises, find a value of \(x\) for which both sides are defined but not equal. $$\cos 1.2 x \cos 0.8 x-\sin 1.2 x \sin 0.8 x=\cos 2 x$$

Step-by-Step Solution

Answer

First, the left side of the equation was rewritten using the difference of two squares. Then, both sides of the equation were graphed in the same viewing rectangle. Whether or not the graphs coincide would determine if the equation is an identity. If the graphs do not coincide, a valid value of \( x \) would be highlighted where the functions are not equal.

Step 1: Rewrite the left side of the equation

You can rewrite the left hand side of the equation using the difference of two squares identity. The identity is \(a^2 - b^2 = (a+b)(a-b)\). Applying it to the equation, you get :\[(\cos 1.2 x + \sin 0.8 x ) (\cos 1.2 x - \sin 0.8 x )\]

Step 2: Graph both sides

Graph both \((\cos 1.2 x + \sin 0.8 x ) (\cos 1.2 x - \sin 0.8 x )\) and \(\cos 2x\) in the same viewing rectangle on a graphing calculator. If the graphs coincide (overlap), then the equation is an identity.

Step 3: Verify the identity

If the graphs coincide, verify your first step to make sure that the left side is the same as the right side. If the graphs do not coincide, identify a value of \( x \) where both functions are defined but not equal.