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Chapter 3: Problem 8

Prove the given identity. $$ \frac{\cos ^{2} \theta}{1+\sin \theta}=1-\sin \theta $$

Short Answer

Expert verified
The identity is proven using the Pythagorean identity and simplification.

Step by step solution

01

- Identity to Prove

The identity to prove is: \[ \frac{\cos ^{2} \theta}{1+\sin \theta}=1-\sin \theta \]
02

- Use Pythagorean Identity

Recall the Pythagorean identity: \[ \cos^2 \theta = 1 - \sin^2 \theta \]
03

- Substitute in Terms of Sine

Replace \( \cos^2 \theta \) in the original equation with \( 1 - \sin^2 \theta \): \[ \frac{1 - \sin^2 \theta}{1 + \sin \theta} = 1 - \sin \theta \]
04

- Simplify the Left Side

Factor the numerator on the left side of the equation: \[ \frac{(1 - \sin \theta)(1 + \sin \theta)}{1 + \sin \theta} = 1 - \sin \theta \]
05

- Cancel the Common Term

Since \(1 + \sin \theta\) appears in both the numerator and the denominator, they can be canceled: \[ 1 - \sin \theta = 1 - \sin \theta \]
06

- Verify Identity

The simplified equation \(1 - \sin \theta = 1 - \sin \theta\) is true, thus proving the original identity.

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Key Concepts

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

Pythagorean Identity
In trigonometry, the Pythagorean identity is a foundational equation, stemming from the Pythagorean theorem. It states that for any angle \theta, the squared sine and cosine values sum up to one: \( \sin^{2} \theta + \cos^{2} \theta = 1 \). This important identity helps in transforming one trigonometric function into another. Here, we used its variation for \cos^{2} \theta. By rearranging the original identity, we get \( \cos^{2} \theta = 1 - \sin^{2} \theta \). This is essential for the first step of our proof.
Trigonometric Simplification
Simplifying trigonometric expressions often involves recognizing patterns and identities to break down complex expressions. In this case, we used the Pythagorean identity to substitute \cos^{2} \theta with \( 1 - \sin^{2} \theta \). Next, in the given exercise, we took our complex fraction \( \frac{1 - \sin^{2} \theta}{1 + \sin \theta} \) and factored the numerator using the difference of squares method. Simplification continues by canceling out identical terms in the numerator and denominator, simplifying the given expression down to a basic form.
Proving Trigonometric Identities
Proving trigonometric identities requires logical steps and the application of known identities and properties. The goal is to transform one side of the equation to look like the other side. Here's a general approach to proving these identities:
  • Identify which identities or properties might be useful.
  • Substitute and manipulate the terms accordingly.
  • Simplify the expression step by step.
  • Ensure both sides of the equation match to verify the identity.
In the exercise, we used the Pythagorean identity, factored terms, and canceled common terms to prove that \( \frac{\text{cos}^{2} \theta}{1 + \sin \theta} = 1 - \sin \theta \). Each step logically follows and simplifies until both sides are identical. This confirms the identity's validity.

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Most popular questions from this chapter

Prove Mollweide's second equation: For any triangle \(\triangle A B C, \frac{a+b}{c}=\frac{\cos \frac{1}{2}(A-B)}{\sin \frac{1}{2} C}\).

Show that each trigonometric function can be put in terms of the sine function.

For any positive angles \(A\), \(B\), and \(C\) such that \(A+B+C=90^{\circ}\), show that $$ \tan A \tan B+\tan B \tan C+\tan C \tan A=1 $$.

Suppose that a point with coordinates \((x, y)=\left(a(\cos \psi-\epsilon), a \sqrt{1-\epsilon^{2}} \sin \psi\right)\) is a distance \(r>0\) from the origin, where \(a>0\) and \(0<\epsilon<1\). Use \(r^{2}=x^{2}+y^{2}\) to show that \(r=a(1-\epsilon \cos \psi)\).11(Note: These coordinates arise in the study of elliptical orbits of planets.)

We showed that \(\sin \theta=\pm \sqrt{1-\cos ^{2} \theta}\) for all \(\theta\). Give an example of an angle \(\theta\) such that \(\sin \theta=-\sqrt{1-\cos ^{2} \theta}\).
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