Question

Graph f and g in the same viewing rectangle. Do the graphs suggest that the equation f(x)=g(x) is an identity? Prove your answer. f(x)=cos² x-sin²x, g(x)=1-2 sin²x

Answer 1

The functions f(x) and g(x), given as cos² x - sin² x and 1 - 2 sin² x, are shown to be equivalent through the use of trigonometric identities, proving f(x) = g(x) is an identity.

Step-by-step explanation:

The student is comparing two trigonometric functions, f(x)=cos² x - sin² x and g(x)=1 - 2 sin² x, to determine if the equation f(x) = g(x) is an identity. We can use the Pythagorean trigonometric identity cos² x + sin² x = 1 and the double-angle formula cos 2x = cos² x - sin² x to prove that f(x) and g(x) are equivalent.

Using the formula for cos 2x, we simplify f(x) as follows:

• f(x) = cos² x - sin² x = cos 2x

Similarly, by substituting 1 - cos² x for sin² x in g(x), we have:

• g(x) = 1 - 2 sin² x = 1 - 2(1 - cos² x) = 2 cos² x - 1 = cos 2x

Both f(x) and g(x) simplify to cos 2x, verifying that f(x) = g(x) is indeed an identity.