Final answer:
The question seeks the value of sin²x, but none of the options provided correspond to the correct Pythagorean identity for sin²x. The exact value can be written as sin²x = (1 - cos 2x)/2, not a simple expression like those given in the options.
Step-by-step explanation:
The question asks for the exact value of sin²x given that you have the sine of x. Referring to the Pythagorean identity, we know that sin²x + cos²x = 1. This can be rearranged to find sin²x by subtracting cos²x from both sides, giving us sin²x = 1 - cos²x. Using the identity cos 2x = cos²x - sin²x, we can also express sin²x as sin²x = (1 - cos 2x)/2. Thus, the exact value of sin²x can be expressed using cos 2x.
If we look at the choices given, option (d) 2cos²x - 1 is the only one that resembles a Pythagorean identity. However, note that this is the formula for cos 2x, not sin²x. Remember that cos 2x = 1 - 2sin²x, which implies that sin²x = (1 - cos 2x)/2. Therefore, none of the choices a) through d) give the exact value of sin²x.