Problem 23 - Algebra 2 and Trigonometry

Question

In \(3-26,\) prove that each equation is an identity. $$ \sin \theta+\frac{\cos ^{2} \theta}{1+\sin \theta}=1 $$

Step-by-Step Solution

Answer

The identity is proven as \(\sin \theta + \frac{\cos^2 \theta}{1 + \sin \theta} = 1\).

Step 1: Analyze the Objective

We need to prove that the given equation is an identity, meaning it holds true for all values of \(\theta\) where it is defined.

Step 2: Simplify the Equation

Start by simplifying the left-hand side (LHS) of the equation: \(\sin \theta + \frac{\cos^2 \theta}{1 + \sin \theta}\). Our goal is to show it equals 1, which is the right-hand side (RHS).

Step 3: Substitute Trigonometric Identity

Recall the Pythagorean identity: \(\cos^2 \theta = 1 - \sin^2 \theta\). Substitute \(\cos^2 \theta\) in the LHS with \(1 - \sin^2 \theta\) to get: \[\sin \theta + \frac{1 - \sin^2 \theta}{1 + \sin \theta}\].

Step 4: Simplify the Fraction

Simplify the fraction: \[\frac{1 - \sin^2 \theta}{1 + \sin \theta} = \frac{(1 - \sin \theta)(1 + \sin \theta)}{1 + \sin \theta} = 1 - \sin \theta\]. The last step assumes \(1 + \sin \theta eq 0\).

Step 5: Combine the Terms

Combine terms in the LHS: \[\sin \theta + (1 - \sin \theta) = 1\]. This shows LHS equals the RHS, thus verifying the identity.

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry, connecting the squares of sine and cosine functions. It is written as:

• \(\sin^2 \theta + \cos^2 \theta = 1\)

This identity emerges from the Pythagorean Theorem applied to the unit circle. It is a valuable tool in simplifying trigonometric expressions, particularly when transforming between sine and cosine.
In the given problem, we utilize the Pythagorean identity to replace \(\cos^2 \theta\) with \(1 - \sin^2 \theta\). This substitution is crucial because it allows us to work with expressions in terms of sine only, making the algebraic manipulation more straightforward.
The beauty of the Pythagorean identity lies in its ability to work for any angle \(\theta\). It is applicable everywhere the trigonometric functions are defined, aiding in various trigonometric proofs and simplifications.

Fraction Simplification

Fraction simplification is a technique used to make equations easier to handle and solve. When faced with a complex fraction, the goal is to reduce it to a simpler form by canceling common factors in the numerator and the denominator.

• In our exercise, we need to simplify \(\frac{1 - \sin^2 \theta}{1 + \sin \theta}\).

Using the identity \(1 - \sin^2 \theta\) allows us to rewrite the fraction as \((1 - \sin \theta)(1 + \sin \theta)\). The term \((1 + \sin \theta)\) appears in both the numerator and the denominator, which enables us to cancel it out under the condition that \(1 + \sin \theta\) is not equal to zero.
This results in the fraction simplifying to \(1 - \sin \theta\), significantly reducing the complexity of the original expression.

Trigonometric Equations

Trigonometric equations, like the one in this exercise, often require proving that two expressions are equivalent under certain conditions. Solving these equations often involves the application of trigonometric identities and simplifications.
To establish the equation as an identity, we start by simplifying the left-hand side (LHS)

• First, we incorporate the Pythagorean identity.
• Next, we simplify fractions to reduce the expression.

In this scenario, simplifying the LHS step by step reveals it equals the right-hand side (RHS), showing \(\sin \theta + \frac{\cos^2 \theta}{1 + \sin \theta} = 1\) for all valid \(\theta\).
Trigonometric equations often require comfort with algebraic manipulation and an understanding of where identities apply. Recognizing patterns and strategically applying identities are keys to mastering these types of problems.