arctanx=sinx

The statement "arctan(x) = sin(x)" is not correct. Here's why:

The function arctan(x), also known as the inverse tangent function, gives the angle whose tangent is x. It returns an angle in radians or degrees, depending on the units used. The domain of arctan(x) is all real numbers (-∞, ∞), and its range is (-π/2, π/2) radians or (-90°, 90°) degrees.

The function sin(x), on the other hand, is the sine function, which gives the ratio of the opposite side to the hypotenuse in a right triangle when the angle is x. The sine function is periodic with a period of 2π radians (or 360 degrees), and its values range between -1 and 1.

These two functions are different and cannot be equated directly. They are related in the sense that the sine of the angle whose tangent is x is equal to the opposite side of the right triangle divided by the adjacent side, which is the same as the value of x. However, this relationship does not imply that arctan(x) = sin(x).

Here's a more accurate relationship between the two functions:

sin(arctan(x)) = x / √(1 + x2)

This equation holds true because arctan(x) gives the angle whose tangent is x, and the sine of that angle will be the ratio of the opposite side to the hypotenuse in the right triangle, which, by definition, is x divided by the square root of the sum of the squares of the opposite and adjacent sides (which is √(1 + x2)).