Problem 93 - Precalculus with Limits

Question

In Exercises 91 - 94, determine whether the statement is true or false. Justify your answer. \( \tan\left(x - \dfrac{\pi}{4}\right) = \dfrac{\tan x + 1}{1 - \tan x} \)

Step-by-Step Solution

Answer

The statement \( \tan\left(x - \dfrac{\pi}{4}\right) = \dfrac{\tan x + 1}{1 - \tan x} \) is false.

Step 1: Identify the equation to prove

The exercise is to prove whether the following equation is true or false: \( \tan\left(x - \frac{\pi}{4}\right) = \frac{\tan x + 1}{1 - \tan x} \).

Step 2: Apply the difference of angles formula to the left side of the equation

The formula for the tangent of the difference of two angles is \(\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A * \tan B}\). Applying this to the left side of the given equation, where A equals x and B equals \(\frac{\pi}{4}\), results in: \( \tan\left(x - \frac{\pi}{4}\right) = \frac{\tan x - \tan\frac{\pi}{4}}{1 + \tan x *\tan\frac{\pi}{4}} \).

Step 3: Replace \(\tan{\pi/4}\) with 1

The value of \(\tan{\pi/4}\) is known to be 1. Hence, replace \(\tan{\pi/4}\) with 1 in the equation to get: \( \tan\left(x - \frac{\pi}{4}\right) = \frac{\tan x - 1}{1 + \tan x * 1} = \frac{\tan x - 1}{\tan x + 1} \).

Step 4: Verify the equality

At this stage, the left side of the equation becomes \(\frac{\tan x - 1}{\tan x + 1}\). However, the original equation to prove was \( \tan\left(x - \frac{\pi}{4}\right) = \frac{\tan x + 1}{1 - \tan x}\). So, the two sides are not equal.

Key Concepts

Tangent Function

The tangent function is one of the six fundamental trigonometric functions. Defined as the ratio of the length of the opposite side to the length of the adjacent side in a right triangle, the tangent is symbolically represented as \( \tan \). It's a periodic function, with a period of \( \pi \), meaning it repeats its values every \( \pi \) radians.

The tangent function is particularly useful because it relates the angles of a triangle to the ratios of its sides. This makes it invaluable when solving problems in trigonometry, physics, engineering, and even in everyday situations like calculating the slope of a ramp. Given an angle \( \theta \), the tangent can be expressed as \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \), where \( \sin \) and \( \cos \) are the sine and cosine functions respectively.

Angle Difference Formula

The angle difference formula for the tangent function is a critical tool in trigonometry. It relates the tangent of the difference between two angles to the tangents of the individual angles themselves. The formula is
\[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \times \tan B} \]
This identity allows us to expand the tangent of an angle expressed as the difference of two other angles, making complex trigonometric calculations more manageable. It is instrumental in proving trigonometric identities and simplifying expressions involving tangents of angle differences.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. These identities are powerful tools used to simplify expressions, solve trigonometric equations, and model periodic phenomena.

There are several fundamental trigonometric identities, including the Pythagorean identities, reciprocal identities, and angle sum and difference identities. Besides the basic equalities, there's a rich tapestry of more complex identities derived from them, which make up the fabric of trigonometric problem-solving strategies.

Proving Trigonometric Identities

Proving trigonometric identities involves verifying that both sides of an equation are identical for all values of the involved angles. The process usually includes a series of algebraic manipulations using known identities, such as the Pythagorean identity or angle sum and difference formulas, to show that the two sides are equivalent.

For the exercise in question, the goal is to determine the truth of the given equation involving the tangent function. To tackle this, one would typically employ the angle difference identity and manipulate the expression algebraically to reach a form that either matches the proposed identity or contradicts it. Proving identities can often lead to deeper insights into the properties and relationships between trigonometric functions.