Question

Identity proofs 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

Question: Prove the identity \(\tan(2\tan^{-1}x) = \frac{2x}{1-x^2}\) and find the values of x for which it holds true. Answer: The identity \(\tan(2\tan^{-1}x) = \frac{2x}{1-x^2}\) is true for all real numbers x except x = 1 and x = -1.

Step-by-step solution

Rewrite the given expression using the double angle formula for tangent

We know that the double angle formula for tangent is given by: $$\tan(2A) = \frac{2\tan A}{1 - \tan^2 A}$$ Apply this formula to the given expression: $$\tan\left(2\tan^{-1}x\right) = \frac{2\tan(\tan^{-1} x)}{1-\tan^2(\tan^{-1} x)}$$

Simplify the expression by canceling inverse functions

As \(\tan(\tan^{-1} x) = x\), we can replace the \(\tan(\tan^{-1} x)\) with x in the expression: $$\tan\left(2\tan^{-1}x\right) = \frac{2x}{1-x^2}$$

Verify the identities

Comparing the simplified expression with the given expression, we see that they are the same: $$\tan \left(2 \tan ^{-1} x\right)=\frac{2 x}{1-x^{2}}$$ Hence, the identity is proved.

Find the domain of the function

To find the valid x values for which the identity holds true, we need to consider the domain of the inverse tangent function: As the domain of the inverse tangent function is all real numbers, we need to find the restrictions for the domain of the tangent function and make sure it satisfies the given identity. In our case, since the denominator shouldn't be equal to zero, we have: $$1 - x^2 \neq 0$$ $$x^2 \neq 1$$ \(x \neq 1\) and \(x \neq -1\) So, the values of x for which the given identity is true are all real numbers except x = 1 and x = -1.

Key concepts

Inverse Tangent Function

The inverse tangent function, often denoted as \( \tan^{-1}(x) \) or arctan(x), is a fundamental concept in trigonometry. It serves as the inverse operation of the tangent function. The inverse tangent function takes a ratio (the result of a tangent operation) and returns the angle whose tangent value is that ratio.

• This function is crucial for converting back from a tangent value to an angle in radians.
• The result of \( \tan^{-1}(x) \) lies within the interval \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), which corresponds to angles in the first and fourth quadrants.
• Understanding \( \tan^{-1}(x) \) helps in solving trigonometric equations where angles are to be determined from given ratios.

In this exercise, the expression \( \tan(2\tan^{-1}(x)) \) is simplified using the property that \( \tan(\tan^{-1}(x)) = x \). This identity is valid for all real numbers, making it a universal tool in trigonometry.

Double Angle Formula

The double angle formula is a key tool in trigonometry that helps simplify expressions involving angles that are twice another angle. Specifically, for the tangent function, the formula is given by:\[\tan(2A) = \frac{2\tan A}{1 - \tan^2 A}\]

• This formula simplifies complex trigonometric problems by reducing them to more manageable expressions.
• It is derived from the angle addition formulas and is essential in proving trigonometric identities.
• In the exercise, substituting \( 2\tan^{-1}(x) \) into this formula helps transform the original expression and verify its identity.

Understanding the double angle formula allows students to tackle problems involving angle manipulation, especially when doubling an angle in trigonometric contexts. It also reinforces the understanding of how trigonometric functions behave in relation to angle changes.

Domain Restrictions

Domain restrictions are important in defining the scope of trigonometric identities and the conditions under which they hold true. In this context, a crucial part of the exercise is identifying where the identity \( \tan(2\tan^{-1}(x)) = \frac{2x}{1-x^2} \) is valid. To do so, one must ensure the expression is defined for all applicable x values.

• The inverse tangent function itself has no restrictions within the real numbers, being defined for all \( x \).
• However, the expression's denominator \( 1 - x^2 \) cannot be zero, creating specific restrictions on \( x \).
• This leads to the condition \( x eq 1 \) and \( x eq -1 \), ensuring that the identity does not result in undefined expressions.

By assessing these domain restrictions and conditions, students can better understand the boundaries within which trigonometric identities are applicable, enhancing their comprehension and application of trigonometric principles in a range of mathematical problems.