Problem 32 - Precalculus Mathematics for Calculus

Question

Prove the identity. $$\tan \left(x-\frac{\pi}{4}\right)=\frac{\tan x-1}{\tan x+1}$$

Step-by-Step Solution

Answer

Identity is proven using the tangent subtraction formula.

Step 1: Recall the Tangent Subtraction Formula

The formula for the tangent of a difference is \( \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b} \). We will apply this to \( \tan \left( x - \frac{\pi}{4} \right) \), where \( a = x \) and \( b = \frac{\pi}{4} \).

Step 2: Substitute the Known Value for \( b \)

Note that \( \tan \left( \frac{\pi}{4} \right) = 1 \). Substituting into the formula, we have \( \tan \left( x - \frac{\pi}{4} \right) = \frac{\tan x - 1}{1 + \tan x \cdot 1} \).

Step 3: Simplify the Expression

The expression becomes \( \frac{\tan x - 1}{1 + \tan x} \). This matches exactly with the right-hand side of the identity \( \frac{\tan x - 1}{\tan x + 1} \).

Step 4: Conclude the Proof

Since both sides of the equation are identical using the tangent subtraction formula, the identity \( \tan \left( x - \frac{\pi}{4} \right) = \frac{\tan x - 1}{\tan x + 1} \) is proven to be true.

Key Concepts

Tangent Subtraction Formula

When dealing with trigonometric identities, the tangent subtraction formula is a crucial tool. It helps us find the tangent of a difference between two angles. The formula is given as:\[\tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}\]This formula is derived from the trigonometric functions and is foundational in understanding how changes in angles affect their tangent values. In our exercise, we specifically look at how this formula is applied when subtracting the angle \(\frac{\pi}{4}\) from \(x\). By setting \(a = x\) and \(b = \frac{\pi}{4}\), and using the known value that \(\tan \frac{\pi}{4} = 1\), we can simplify and prove the given identity.
Understanding the tangent subtraction formula allows students to manipulate expressions involving tangent functions and prove various identities. It's a powerful technique to use in algebraic manipulations within trigonometry.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, particularly in examining the relationships between the angles and sides of triangles. These functions include sine, cosine, and tangent, among others.The tangent function, abbreviated as \(\tan\), is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. Specifically for our exercise, we focus on tangent and its properties.
Key points about tangent include:

• The tangent function is periodic with a period of \(\pi\).
• Tangent is undefined at odd multiples of \(\frac{\pi}{2}\) due to division by zero in its ratio form.
• Unlike sine and cosine, the tangent function has asymptotes, lines where the function approaches but never touches.

In our specific identity, understanding these properties helps us apply the tangent subtraction inside broader trigonometric manipulations. By grasping the fundamental behavior of tangent, students can better approach not only this identity proof but also similar trigonometric equations and expressions.

Proof Techniques

Proof techniques are essential when verifying mathematical identities or solving complex equations. Developing these skills allows you to demonstrate why specific equations are always true or identify contradictions in claims.For the given tangent identity, the primary proof technique used is substitution and simplification. By recognizing known values (such as \(\tan \frac{\pi}{4} = 1\)), and substituting these into the tangent subtraction formula, we simplify the expression step-by-step.
Some additional proof strategies include:

• Direct Proof: Uses logical sequences and known truths to derive a conclusion directly.
• Contradiction: Assumes that the statement you want to prove is false, and shows that this assumption leads to an illogical conclusion.
• Induction: Proves a base case and assumes true for one iteration to prove for the subsequent iteration.

By using proof techniques such as substitution and simplification, students can gain confidence in tackling complex problems by breaking them down into smaller, more manageable parts. With practice, proof strategies enhance understanding and mastery of mathematics.