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Chapter 29: Problem 960

Prove that the following equation is an identity: \((1-\sin x) /(\cos x)=(\cos x) /(1+\sin x)\).

Short Answer

Expert verified
To prove the identity \(\frac{1 - \sin x}{\cos x} = \frac{\cos x}{1 + \sin x}\), we first cross-multiply and obtain \((1 - \sin x)(1 + \sin x) = \cos^2 x\). Next, using the difference of squares formula, we get \(1 - \sin^2 x=\cos^2 x\). Using the Pythagorean identity, we observe that both sides of the equation are equal, proving the identity.

Step by step solution

01

Write down the given equation

First, let's write down the given equation: \(\frac{1 - \sin x}{\cos x} = \frac{\cos x}{1 + \sin x}\)
02

Cross-multiply

In this step, we will perform cross-multiplication to eliminate the denominators. Multiply both sides of the equation by \(\cos x\ (1 + \sin x)\): \((1 - \sin x) (1 + \sin x) = \cos^2 x\)
03

Use the difference of squares formula

Now, we can simplify the left-hand side (LHS) of the equation by using the difference of squares formula. The formula is: \((a - b)(a + b) = a^2 - b^2\) In our case, a = 1 and b = \(\sin x\): \((1 - \sin x) (1 + \sin x) = 1^2 - (\sin x)^2\) So the LHS of the equation becomes: \(1 - \sin^2 x = \cos^2 x\)
04

Use the Pythagorean identity

We can use the Pythagorean identity which states that: \(\sin^2 x + \cos^2 x = 1\) We can rearrange this identity to express \(\cos^2 x\): \(\cos^2 x = 1 - \sin^2 x\)
05

Compare both sides

Now that we have obtained \(\cos^2 x\), let's compare both sides: LHS: \(1 - \sin^2 x\) RHS: \(\cos^2 x = 1 - \sin^2 x\) Both sides are equal, which means that we have proved the identity. The given equation is indeed an identity, as the left-hand side is equal to the right-hand side for any value of x.

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Key Concepts

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

Difference of Squares
In algebra, the difference of squares is a technique used to simplify expressions. This method involves rewriting a product of two binomials with a special pattern. It states that
  • \((a - b)(a + b) = a^2 - b^2\)
  • This pattern occurs because the middle terms cancel each other out.
When applied to trigonometric functions, like in the given exercise, it allows us to simplify complex expressions more easily. Here, we use the difference of squares on the left-hand side,
  • where \(a = 1\) and \(b = \sin x\),
  • resulting in the simplified form \(1 - \sin^2 x\).
This simplification directly helps us to link the expression with Pythagorean identity, allowing for further evaluation. Understanding this concept is critical for working through many trigonometric identities.
Cross-Multiplication
Cross-multiplication is a simple method often used to solve equations involving fractions. When you have an equation in the form
  • \(\frac{a}{b} = \frac{c}{d}\),
  • cross-multiplying involves multiplying across the equal sign so that \((a \cdot d = b \cdot c)\).
This technique effectively eliminates the denominators making the equation easier to solve.
  • In our problem, we started with \(\frac{1 - \sin x}{\cos x} = \frac{\cos x}{1 + \sin x}\),
  • which was cross-multiplied to \((1 - \sin x)(1 + \sin x) = \cos^2 x\) for simplification.
Employing cross-multiplication is essential especially when dealing with identities and inequalities in trigonometry.
Pythagorean Identity
The Pythagorean identity is a fundamental property within trigonometry that expresses a relationship between
  • sine and cosine.
It is stated as \(\sin^2 x + \cos^2 x = 1\). This identity allows one to express one trigonometric function in terms of another, enabling simplifications and proofs.
  • In our exercise, the rearranged identity \(\cos^2 x = 1 - \sin^2 x\) was essential to show that both sides of the equation were equal.
  • This shows how the difference of squares converges to the Pythagorean identity.
This concept is a cornerstone of trigonometry and is crucial for proving the identity in question.
Trigonometric Simplification
Trigonometric simplification is the process of making trigonometric expressions more manageable or simpler. This involves applying identities like the Pythagorean identity, or techniques such as the difference of squares and cross-multiplication.
  • Simplifying the given equation using these techniques led to the final proof that the equation is a true identity.
  • Simplification helps when solving trigonometric equations, as it often makes apparent relationships that were not immediately obvious.
Having a good grasp of these basic simplification methods is imperative for students learning trigonometry as it improves problem-solving skills and understanding of how different identities connect.

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