Question

Write the expression in algebraic form. $$ \tan \left(\tan ^{-1} x\right) $$

Short answer

The expression \(\tan(\tan^{-1}(x))\) simplifies to its algebraic form as \(x\).

Step-by-step solution

Relationship between tangent and arctangent functions

The algebraic relationship between the tangent function and its inverse is given by: \(x = \tan(\tan^{-1}(x))\) This means that the tangent function applied to its inverse is simply equal to the input \(x\), which is the algebraic expression.

Simplify the given expression

Using the relationship from step 1, we can now simplify the expression \(\tan(\tan^{-1}(x))\): \(\tan(\tan^{-1}(x)) = x\) So, the given expression simplifies to its algebraic form as: \(x\)

Key concepts

Tangent and Arctangent Relationship

Understanding the tangent and arctangent relationship is crucial when dealing with trigonometric expressions. Essentially, the arctangent function, denoted as \(\tan^{-1}(x)\), is the inverse of the tangent function. It is used to determine an angle whose tangent value is \(x\). One of the key aspects of any inverse function is that when you apply the original function to it, you should obtain the initial input.

For instance, if you have an angle whose tangent is \(x\), and you apply the arctangent function to \(x\), you'll get the original angle back. Similarly, when you apply the tangent function to this angle, the result is \(x\). Mathematicians denote this as \(\tan(\tan^{-1}(x)) = x\). This relationship is a direct consequence of the definition of inverse functions and is fundamental in simplifying expressions involving the tangent and arctangent functions. This principle is not only restricted to tangent and arctangent functions but applies to all pairs of inverse trigonometric functions.

Algebraic Simplification

Algebraic simplification involves rewriting expressions into a simpler or more understandable form. The goal of simplification is often to make the expressions more comprehensible or to prepare them for solving equations, evaluating values, or performing calculus operations. One typical example of simplification is when you are presented with an expression such as \(\tan(\tan^{-1}(x))\).

Using the knowledge about inverse functions, we understand that this expression simplifies directly to \(x\), reducing the trigonometric function to its algebraic form. This process of replacement is a key algebraic operation, saving time and helping to illuminate the structure of more complex expressions. Simplifying expressions is a fundamental skill in algebra that aids students in solving problems across various fields of mathematics.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables where both sides of the equation are defined. These identities are extremely helpful in solving trigonometric equations, simplifying expressions, and even in calculus for integrals and derivatives of trigonometric functions.

There are several basic trigonometric identities such as reciprocal identities, Pythagorean identities, angle sum and difference identities, double angle identities, etc. For instance, the Pythagorean identities relate the squares of the sine and cosine functions to 1, which can be written as \(\sin^2(x) + \cos^2(x) = 1\).

Understanding these identities allows you to simplify complex trigonometric expressions, convert between different trigonometric functions, and find the values of unknown variables. Remembering these identities and knowing when to apply them can simplify seemingly difficult problems to more manageable ones.

Calculus

Calculus, the mathematical study of continuous change, is divided primarily into two branches: differential calculus and integral calculus. Differential calculus deals with the concept of the derivative, which represents the rate of change of a function with respect to a variable. Integral calculus, on the other hand, involves the integral, which represents the area under a curve or the accumulation of quantities.

The connection between trigonometric functions and calculus is profound as they are frequently used to model periodic phenomena. With tools like derivatives and integrals, you can solve problems involving rates of changes (like velocity), areas, and volumes where trigonometric functions play a role. For trigonometric expressions, such as \(\tan(\tan^{-1}(x))\), calculus allows us to not only simplify these expressions but also to understand the behaviors of these functions graphically and the relationship they share with other mathematical concepts.