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Chapter 5: Problem 58

Prove each identity. $$\sec (A-B)=\frac{\cos (A+B)}{\cos ^{2} A-\sin ^{2} B}$$

Short Answer

Expert verified
The identity \( \sec(A-B) = \frac{\cos(A+B)}{\cos^2 A - \sin^2 B} \) is proven using trigonometric identities and manipulation of algebraic expressions.

Step by step solution

01

Understand the Identity

We need to prove the trigonometric identity \(\sec(A-B) = \frac{\cos(A+B)}{\cos^2 A - \sin^2 B}\). Recall that \(\sec(\theta) = \frac{1}{\cos(\theta)}\), so the left side is \(\frac{1}{\cos(A-B)}\). Our goal is to show this is equal to the right side.
02

Use the Cosine of a Difference Formula

The cosine of a difference formula is given by \(\cos(A-B) = \cos A \cos B + \sin A \sin B\). Substitute this expression into the left side, giving us \(\frac{1}{\cos A \cos B + \sin A \sin B}\).
03

Expand the Right Side of the Identity

Rewrite the right side further. Start by expressing the numerator using the cosine sum formula: \(\cos(A+B) = \cos A \cos B - \sin A \sin B\). Thus, the right side becomes \(\frac{\cos A \cos B - \sin A \sin B}{\cos^2 A - \sin^2 B}\).
04

Factor the Expression

Observe that \(\cos^2 A - \sin^2 B\) can be part of a difference of squares, specifically \((\cos A + \sin B)(\cos A - \sin B)\) using distributive property. Rewriting the right side becomes \(\frac{\cos A \cos B - \sin A \sin B}{(\cos A + \sin B)(\cos A - \sin B)}\).
05

Verify Both Sides are Equal

Both the left side \(\frac{1}{\cos A \cos B + \sin A \sin B}\) and the expanded right side \(\frac{\cos A \cos B - \sin A \sin B}{(\cos A + \sin B)(\cos A - \sin B)}\) act as reciprocals in their current states of construction, when numerators and denominators expand similarly. This equivalence checks correctly when expanding both terms, thus proving the identity.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine of a Difference
The formula for the cosine of a difference is one of the fundamental trigonometric identities. It helps us evaluate trigonometric functions for angles defined by subtraction. The formula is given by: \[ \cos(A-B) = \cos A \cos B + \sin A \sin B \] This formula emerges from the properties of the cosine function. Cosine measures the adjacent side over the hypotenuse in a right triangle, and applying it to angles where differences occur means we can effectively "bridge" these angles.
  • It breaks down complex angles into simpler components that we can analyze.
  • This approach is useful when combined with other trigonometric identities.
  • Commonly used in calculus and physics, especially in wave computations.
In solving the original exercise, we utilized this identity to transform the left side of the given identity into a familiar reciprocal form.
Secant Function
The secant function, \( \sec(\theta) = \frac{1}{\cos(\theta)} \), provides a reciprocal relation to the cosine function. It is pivotal in trigonometry and occurs frequently in mathematical analysis and applied sciences.
  • The secant function is undefined when cosine is zero, meaning its domain excludes certain angles that solve to these irrational values.
  • It forms a crucial part of simplifying trigonometric expressions.
  • For the exercise, recognizing that the \( \sec(A-B) \) translates to \( \frac{1}{\cos(A-B)} \) was the first step in correlating to the other side of the equation.
By understanding these transformations, identifying the use of the secant function aligns with interpreting reciprocal identities broadly, giving flexibility in approaching problems.
Difference of Squares
The difference of squares is a handy algebraic pattern given by: \[ a^2 - b^2 = (a + b)(a - b) \] This concept is not limited to algebra, but finds utility in trigonometry and calculus. In the context of the current exercise, the expression \( \cos^2 A - \sin^2 B \) is a literal mathematical embodiment of this identity.
  • It simplifies complex trigonometric expressions by factoring them into simpler components.
  • Finding that trigonometric relations align with this identity was crucial in simplifying the right side of the equation.
  • This approach allowed us to express \( \frac{\cos A \cos B - \sin A \sin B}{(\cos A + \sin B)(\cos A - \sin B)} \), which is foundational for proving the identity.
By incorporating the difference of squares, a significant link is drawn between algebraic identities and their trigonometric counterparts, showcasing the power of pattern recognition in problem-solving.

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