Question

In Exercises 117-126, write the expression in algebraic form. \(\tan (\arctan x)\)

Short answer

The algebraic form of the expression \(\tan (\arctan x)\) is 'x'

Step-by-step solution

Understand the expression

The problem presents the expression \(\tan (\arctan x)\). Here the functions \(\tan\) and \(\arctan\) are the tangent and inverse tangent functions respectively. The function \(\tan\) takes the angle and gives the tangent of it, and \(\arctan\) does the opposite, it takes the tangent of an angle and gives the original angle.

Applying inverse trigonometric identities

We know from inverse trigonometric identities that applying a trigonometric function and its inverse consecutively nullifies each other. That is, a function followed by its inverse will result in the original value, and inversely, an inverse function followed by its function also results in the original value. So for any real number 'x', \(\tan (\arctan x)\) will equal 'x'

Final simplification

According to the inverse trigonometric identities mentioned in step 2, we can simplify the given expression \(\tan (\arctan x)\) simply as 'x' as each function undoes the effect of the other.