Final answer:
To solve the equation tan(x)=1 within the interval [-π/2, π/2], we identify x as π/4 radians (or 45 degrees), since this is the angle where the tangent has a value of 1.
Step-by-step explanation:
To solve the equation tan(x)=1 on the interval [-π/2, π/2], we need to determine the angle x for which the tangent function has a value of 1. The tangent of an angle in a right-angled triangle is the ratio of the side opposite to the angle to the side adjacent to the angle. Since we know that tan(x) = 1, we are looking for an angle where the opposite and adjacent sides are equal, which is the case for a 45-degree angle, or π/4 radians.
However, because the tangent function is periodic, we must consider the interval we are restricted to. On the interval [-π/2, π/2], which is the same as [-90 degrees, 90 degrees], the function tan(x) has a period of π, so it will repeat every π radians. But within this specified interval, there is only one angle for which tan(x) equals 1, and that is x = π/4 radians (or 45 degrees).
Thus, the solution to the equation within the given interval is x = π/4.