Problem 64 - Algebra and Trigonometry with Analytic Geometry

Question

Verify the identity by transforming the lefthand side into the right-hand side. $$\frac{\sin \theta+\cos \theta}{\cos \theta}=1+\tan \theta$$

Step-by-Step Solution

Answer

The identity is verified as both sides simplify to \(1 + \tan \theta\).

Step 1: Simplify the Left-Hand Side

The left-hand side (LHS) of the given identity is \( \frac{\sin \theta + \cos \theta}{\cos \theta} \). Begin by splitting the fraction: \[ \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\cos \theta}. \] This simplifies to \( \tan \theta + 1 \).

Step 2: Compare with Right-Hand Side

Now, compare the simplified LHS \( \tan \theta + 1 \) with the RHS of the identity, which is \( 1 + \tan \theta \). Both expressions are identical because addition is commutative. It confirms that the two sides are equal.

Key Concepts

Simplifying Expressions

In mathematics, simplifying expressions is about making them easier to work with. This often means reducing an expression to its simplest form, where it's more meaningful.
To simplify \[\frac{\sin \theta + \cos \theta}{\cos \theta}\]one effective strategy is using basic arithmetic or known identities.
First, you separate the terms into smaller fractions: \[\frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\cos \theta}\]This is key because each term becomes easier to manage:

• \(\frac{\sin \theta}{\cos \theta}\) transforms to \(\tan \theta\), the tangent function.
• \(\frac{\cos \theta}{\cos \theta}\), simplifies to 1, due to any number divided by itself being 1.

When combining these cleaned-up parts, you achieve the simplified expression \(\tan \theta + 1\).
Expressing trigonometric identities in simpler forms not only makes them more accessible but also clearer when verifying or solving equations.

Commutative Property

The commutative property is a fundamental rule in mathematics that helps simplify calculations and expressions. The property states that the order in which two numbers are added or multiplied does not affect the result.
In this identity verification, we use the commutative property of addition. It tells us that \(a + b\) is equal to \(b + a\).

• For our solved expression \(\tan \theta + 1\), you can rearrange it to become \(1 + \tan \theta\).
• This reordering shows the flexibility of mathematical structures without altering the values.

The commutative property is crucial in verifying identities because it confirms that different appearances of an expression can truly be the same, simply written in another order.

Verifying Identities

Verifying trigonometric identities is about proving that two trigonometric expressions are equivalent, even if they look different. It's an essential skill in trigonometry, used to confirm results and simplify complex problems.
The process involves transforming one side of the identity to match the other. For our problem, this meant simplifying the left-hand side \(\frac{\sin \theta + \cos \theta}{\cos \theta}\) and checking if it equals the right-hand side \(1 + \tan \theta\).
Here's a structured method to verify identities:

• Choose one side to simplify: Usually the more complex side.
• Use algebraic manipulations: Such as factoring, distributing, or using known identities.
• Apply trigonometric identities: This might involve definitions or properties of functions like sine, cosine, or tangent.
• Compare the results: After simplification, ensure both sides are identical or can be shown as equivalent.
• Utilize properties: Like the commutative property when necessary to reorder terms.

Verifying identities not only enhances your mathematical understanding but also proves the relationship within trigonometric functions, strengthening problem-solving skills.