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**Hint:**

Arc functions are also termed as arcus functions, anti-trigonometric functions, inverse trigonometric functions or cyclometric functions. The inverse trigonometry functions have major applications in the field of engineering, physics, geometry and navigation.

There are particularly six inverse trig functions for each trigonometric ratio. The inverse of six important trigonometric functions are:

Arcsine

Arccosine

Arctangent

Arc cotangent

Arc secant

Arc cosecant

Arc of some trigonometric function is the same as the inverse of the function. Arc prefix is commonly used to name the trigonometric functions. For example, \[\arcsin x\] = \[\dfrac{1}{2}\]means the angle whose sine is making \[\dfrac{1}{2}\]. Thus, we can say that \[{{\sin }^{-1}}(x)\]= \[\dfrac{1}{2}\] or x = \[{{30}^{\circ }}\].

To solve this question, find the range of all three trigonometric functions and add them to find the right answer.

**Complete step by step solution:**

Step 1: Let us first find the range of those three trigonometric functions separately.

Since, we know \[\arcsin x+\arccos x\] = \[\dfrac{\pi }{2}\]

Therefore, range of \[\arctan x\] is (\[-\dfrac{\pi }{2}\], \[\dfrac{\pi }{2}\])

Step 2: Minimum value of \[\arcsin x+\arccos x+\arctan x\] = \[\dfrac{\pi }{2}\] + (\[-\dfrac{\pi }{2}\]) = 0

Maximum value of \[\arcsin x+\arccos x+\arctan x\] = \[\dfrac{\pi }{2}\]+\[\dfrac{\pi }{2}\] = \[\pi \]

Hence, the range of \[\arcsin x+\arccos x+\arctan x\] = (0, \[\pi \])

**Note:**

To solve this question, the students should be familiar with trigonometric functions. The basics of trigonometry values are a must. Try to visualise such questions through graphs that can perform such trigonometric functionalities. Once you get familiar with trigonometric graphs and functions you can easily find the values of inverse trigonometric functions. Also, for range, observe the round brackets () are used to write the range as the values 0 and \[\pi \]are not included since the range of \[\arctan x\] is (\[-\dfrac{\pi }{2}\], \[\dfrac{\pi }{2}\])

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