Principal Value of (tan^-1) 1

To find the principal value of (tan^-1) 1, we need to determine the angle whose tangent is equal to 1. In other words, we are looking for the angle θ such that tan(θ) = 1.

Finding the Angle

We know that the tangent of an angle is equal to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle in a right triangle. In this case, we have:

tan(θ) = 1

This means that the side opposite the angle is equal to the side adjacent to the angle. We can construct a right triangle with these side lengths. Let's call the length of both sides x.

Constructing the Right Triangle

By using the Pythagorean theorem, we can find the length of the hypotenuse. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

x^2 + x^2 = c^2

2x^2 = c^2

Taking the square root of both sides, we get:

√(2x^2) = √(c^2)

√2x = c

Calculating the Angle

Now that we have the lengths of all three sides of the right triangle, we can calculate the angle θ. We know that the tangent of an angle is equal to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. In this case, we have:

tan(θ) = x/x = 1

Therefore, θ = tan^-1(1)

Principal Value

The principal value of an inverse trigonometric function is the value within a certain range that is chosen as the "principal" or main value. For the tangent function, the principal value is typically chosen within the range (-π/2, π/2), which means -π/2 < θ="" />< />

In this case, the angle whose tangent is equal to 1 lies within this range. Therefore, the principal value of (tan^-1) 1 is π/4.

Hence, the correct answer is option 'D' (π/4).