Question

Prove the following identities and give the values of \(x\) for which they are true. $$\tan \left(2 \tan ^{-1} x\right)=\frac{2 x}{1-x^{2}}$$

Short answer

Based on the given solution, the identity $$\tan\left(2\tan^{-1}(x)\right) = \frac{2x}{1-x^2}$$ is true for all values of \(x\) except \(x = \pm 1\).

Step-by-step solution

Express Left-Hand Side in Terms of Sine and Cosine Functions

To simplify the left-hand side of the equation, use the double angle formula for tangent: $$\tan(2\theta) = \frac{2\tan(\theta)}{1-\tan^2(\theta)}$$ In this case, let \(\theta = \tan^{-1}(x)\). Then the left-hand side becomes: $$\tan\left(2\tan^{-1}(x)\right) = \frac{2\tan(\tan^{-1}(x))}{1-\tan^2(\tan^{-1}(x))}$$

Simplify the Expression

Using the fact that \(\tan(\tan^{-1}(x)) = x\), we can further simplify the expression: $$\frac{2\tan(\tan^{-1}(x))}{1-\tan^2(\tan^{-1}(x))} = \frac{2x}{1-x^2}$$

Compare with Right-Hand Side and Find the Values of \(x\)

Now, we compare the simplified expression with the right-hand side of the equation, which is also \(\frac{2x}{1-x^2}\). Since both sides are equal, it is shown that the identity is true. The identity holds true for all values of \(x\) except when the denominator becomes zero. So, we find the values of \(x\) for which this occurs: $$1 - x^2 = 0$$ Solving for \(x\), we find \(x = \pm 1\). So, the identity is true for all \(x\) except \(x = \pm 1\).

Key concepts

Double Angle Formulas

Double angle formulas are key tools in trigonometry that help relate expressions involving double angles to single ones. For the tangent function, the double angle formula is given as \[\tan(2\theta) = \frac{2 \tan(\theta)}{1 - \tan^2(\theta)}\]This formula allows us to express \( \tan(2\theta) \) in terms of \( \tan(\theta) \) without directly doubling the angle measure.
To understand this formula, let's break it down:

• The numerator, \(2 \tan(\theta)\), simply doubles the tangent's value.
• The denominator, \(1 - \tan^2(\theta)\), corrects for the range of the tangent function, which can potentially reach infinity (as tangent has vertical asymptotes).

By using the double angle formula, you can transform expressions involving angles like \( 2\tan^{-1}(x) \) into more manageable algebraic forms. This approach makes calculations and the proving of identities much simpler.

Inverse Trigonometric Functions

Inverse trigonometric functions are the reverse operations of standard trigonometric functions. Specifically, they are used to find angles when their trigonometric values are known. The inverse tangent, written as \( \tan^{-1}(x) \) or sometimes as \( \arctan(x) \), finds the angle whose tangent value is \( x \).
These functions are particularly useful in calculus and geometry when dealing with angle measures that are not already known. For instance, if you have \( x \) such that \( \tan(\theta) = x \), then \( \theta = \tan^{-1}(x) \).
Important properties of inverse trigonometric functions:

• They are multivalued, meaning that an infinite number of values can satisfy the function, but the principal value is taken to be within defined limits; for \( \tan^{-1}(x) \), the range is usually \(-\frac{\pi}{2} < \theta < \frac{\pi}{2}\).
• Inverse functions convert trigonometric ratios back to angles, which can be used to explore complex trigonometric identities.

Understanding how to maneuver between functions and their inverses is crucial in solving trigonometric identities and equations.

Tangent Function

The tangent function is a fundamental trigonometric function that describes the ratio of the opposite to the adjacent side of a right triangle. It is one of the primary functions in trigonometry besides sine and cosine. Notably, \[\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}\]This ratio becomes a critical piece in many mathematical calculations, especially when analyzing wave patterns, oscillations, and rotations.
Key characteristics of the tangent function include:

• It is periodic with a period of \(\pi\), which means that \(\tan(\theta + \pi) = \tan(\theta)\).
• Unlike sine and cosine, tangent has no maximum or minimum values; it is unbounded and can take any real value.
• The function has vertical asymptotes at odd multiples of \(\frac{\pi}{2}\), where it is undefined.

Understanding the behavior of the tangent function is vital for using it effectively in expressions and equations like the identity \( \tan(2\tan^{-1}(x)) = \frac{2x}{1-x^2} \). By mastering the behavior and properties of tangent, solving complex trigonometric problems becomes attainable.