Problem 30 - Algebra and Trigonometry

Question

\(23-40\) . Prove the identity. $$ \tan \left(x-\frac{\pi}{4}\right)=\frac{\tan x-1}{\tan x+1} $$

Step-by-Step Solution

Answer

The identity is proved using the tangent of a difference formula.

Step 1: Recall Trigonometric Identity for Tangent of a Difference

To solve the expression, recall the trigonometric identity for the tangent of a difference:\[\tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}\]Here, let \(a = x\) and \(b = \frac{\pi}{4}\).

Step 2: Calculate \(\tan\left(\frac{\pi}{4}\right)\)

Using known values of tangent for special angles, we have:\[\tan\left(\frac{\pi}{4}\right) = 1\]

Step 3: Substitute Values into the Identity

Substitute \(a = x\) and \(\tan\left(\frac{\pi}{4}\right) = 1\) into the tangent difference identity:\[\tan\left(x - \frac{\pi}{4}\right) = \frac{\tan x - 1}{1 + \tan x \cdot 1}\]Simplifying gives:\[\tan\left(x - \frac{\pi}{4}\right) = \frac{\tan x - 1}{1 + \tan x}\]

Step 4: Simplify the Expression to Match the Given Formula

The equation \(\tan\left(x - \frac{\pi}{4}\right) = \frac{\tan x - 1}{\tan x + 1}\) matches the derived formula. This confirms the identity.The identity given in the problem matches what we derived using the tangent difference identity.

Key Concepts

Understanding the Tangent of a Difference

The tangent of a difference identity is a useful tool in trigonometry. It's employed to simplify expressions involving the tangent function. In general, it expresses the tangent of the difference between two angles \(a\) and \(b\): \[\tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}\]This expression lets us rewrite complex tangent expressions into a simpler form. The choice of angles \(a\) and \(b\) depends on the specific problem, such as with this exercise where we have \(a = x\) and \(b = \frac{\pi}{4}\). Understanding this identity allows students to

• Break down more complex trigonometric expressions, making them easier to analyze and solve
• Demonstrate the relationship between different trigonometric functions
• Provide a method to not only solve problems but also to check the work they've done

In this problem, we know the goal is to prove a given identity, so the efficient application of the tangent of a difference identity is critical.

The Role of Special Angles in Trigonometry

Special angles, such as \(\frac{\pi}{4}\), are crucial in trigonometry because they have well-known trigonometric values. These values help simplify expressions effectively.For the angle \(\frac{\pi}{4}\):

• The sine and cosine values are \(\sin\left(\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\).
• The tangent value becomes \(\tan\left(\frac{\pi}{4}\right) = 1\) because \(\tan\theta = \frac{\sin\theta}{\cos\theta}\).

Understanding these specific values allows students to substitute known quantities into identities and expressions smoothly.Such substitutions are not only helpful but necessary when verifying identities or working through problems involving known angle measures. These special angle values accelerate the process of simplification, leading to quicker and often more accurate results.

Techniques for Simplifying Trigonometric Expressions

Simplifying expressions in mathematics, particularly trigonometric ones, requires a strategic approach. This exercise involves simplifying a tangent expression after applying a trigonometric identity.Here are some strategies:

• Use known identities, like the tangent of a difference, as foundation steps. They provide the structure for your simplification.
• Substitute the specific values of trigonometric functions for special angles at every possible opportunity, just like \(\tan\left(\frac{\pi}{4}\right) = 1\) in this problem.
• After substitution, focus on cancelling terms and reducing fractions wherever possible. This often requires combining like terms or factoring.

In the exercise, by substituting known values and applying fraction simplification, you find that both sides of the identity are equivalent.This approach not only proves the identity but also reinforces the relationship between different trigonometric functions. It enables students to solve problems with confidence.