Problem 38 - Precalculus Mathematics for Calculus

Question

Prove the identity. $$1-\tan x \tan y=\frac{\cos (x+y)}{\cos x \cos y}$$

Step-by-Step Solution

Answer

The identity is true; both sides equal after simplification.

Step 1: Recall Trigonometric Identities

We need to use some basic trigonometric identities:1. The tangent addition formula: \( \tan(x+y) = \frac{\tan x + \tan y}{1 - \tan x \tan y} \).2. The cosine addition formula: \( \cos(x+y) = \cos x \cos y - \sin x \sin y \).We will use these identities in our steps to verify the given identity.

Step 2: Simplify the Right-Hand Side

The given identity is:\(1 - \tan x \tan y = \frac{\cos(x+y)}{\cos x \cos y}\).Let's manipulate the right-hand side (RHS):Using the cosine addition formula:\[\cos(x+y) = \cos x \cos y - \sin x \sin y\]Thus, the RHS becomes:\[\frac{\cos x \cos y - \sin x \sin y}{\cos x \cos y}\]

Step 3: Break Down the Fraction

When breaking down the fraction on the RHS:\[\frac{\cos x \cos y - \sin x \sin y}{\cos x \cos y} = \frac{\cos x \cos y}{\cos x \cos y} - \frac{\sin x \sin y}{\cos x \cos y}\]This simplifies to:\[1 - \frac{\sin x \sin y}{\cos x \cos y}\]Thus, the RHS becomes similar to \(1 - \tan x \tan y\).

Step 4: Use Tangent Definition

Recall the definition for tangent: \(\tan x = \frac{\sin x}{\cos x}\) and \(\tan y = \frac{\sin y}{\cos y}\).Therefore:\[\tan x \tan y = \left(\frac{\sin x}{\cos x}\right) \left(\frac{\sin y}{\cos y}\right) = \frac{\sin x \sin y}{\cos x \cos y}\]Replacing this back in the earlier fraction we obtained:\[1 - \tan x \tan y = 1 - \frac{\sin x \sin y}{\cos x \cos y}\]This confirms that \(1 - \tan x \tan y\) is indeed equal to \(\frac{\cos(x+y)}{\cos x \cos y}\).

Step 5: Conclusion

Both sides of the given identity simplify to the same form, confirming the identity is true. Thus, we have proved that:\[1 - \tan x \tan y = \frac{\cos(x+y)}{\cos x \cos y}\]

Key Concepts

Understanding the Tangent Addition Formula

The tangent addition formula is a key trigonometric identity that helps us compute the tangent of the sum of two angles. Let's dive into how it works and where it's useful.

The formula is given by: \( \tan(x+y) = \frac{\tan x + \tan y}{1 - \tan x \tan y} \). This shows that to find the tangent of the sum, you need to know the tangents of the individual angles and adjust them using a specific mechanism.

Why is this formula handy?

• It simplifies calculating the tangent for complex angles, avoiding direct computations which can be cumbersome.
• It helps in verifying identities, like in our original exercise, enabling the transformation and simplification of expressions involving tangents.

Furthermore, understanding the tangent addition formula is crucial for solving problems in calculus and physics where trigonometric functions regularly appear. This formula showcases the interconnectedness of trigonometric functions.

Decoding the Cosine Addition Formula

The cosine addition formula is another foundational identity in trigonometry. It expands the cosine of the sum of two angles into a form that's easier to work with. Let's see how this works.

The formula is: \( \cos(x+y) = \cos x \cos y - \sin x \sin y \). This reveals that cosine can be broken down into differences of products of cosines and sines of individual angles.

This formula is immensely useful because:

• It aids in simplifying cosine expressions, making them more manageable in equations.
• It is crucial in proving intricate trigonometric identities, as it shows relationships between different trigonometric functions.

In our exercise, using the cosine addition formula to transform the right-hand side played a vital role in proving the identity. It serves as an essential tool both in academic problems and real-life applications such as signal processing and wave analysis.

Exploring Trigonometric Proofs

Trigonometric proofs are logical reasoning exercises where you show one trigonometric expression equals another. They require a strong grasp of identities and the ability to manipulate expressions skillfully.

Proving the identity \(1 - \tan x \tan y = \frac{\cos(x+y)}{\cos x \cos y}\) involves:

• Using known identities like the tangent and cosine addition formulas.
• Simplifying expressions via algebraic manipulation to make both sides match.

Why are proofs important? They develop critical thinking and deepen understanding of mathematics. They showcase the coherence of mathematical systems, confirming that different representations of functions are actually equivalent.

Understanding trigonometric proofs ensures a robust comprehension of trigonometry, preparing you for advanced studies involving calculus, engineering, and physics. They demonstrate the elegance and power of mathematical reasoning, showing how complex expressions can transform and reduce to simpler forms.