Problem 52 - Algebra 2

Question

Verify each identity. Express \(\frac{\cos \theta}{\sec \theta+\tan \theta}\) in terms of \(\sin \theta\)

Step-by-Step Solution

Answer

The identity \(\frac{\cos \theta}{\sec \theta+\tan \theta}\) simplifies to \(\frac{1 - \sin^2 \theta}{1+\sin \theta}\) when expressed in terms of the \(\sin \theta\).

Step 1: Substitute fundamental identities

Begin with the given identity and substitute the corresponding fundamental identities for \(\sec \theta\) and \(\tan \theta\). It is known that \(\sec \theta = \frac{1}{\cos \theta}\) and \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). After substitution, we get: \(\frac{\cos \theta}{\frac{1}{\cos \theta}+\frac{\sin \theta}{\cos \theta}}\).

Step 2: Simplify identities

Next step is to simplify the equation by performing the addition in the denominator and to get rid of the fraction. We do this by multiplying top and bottom by \(\cos \theta\). This gives: \(\frac{\cos^2 \theta}{1+\sin \theta}\).

Step 3: Substitute Pythagorean Identity

Finally, substitute the Pythagorean identity \(\cos^2 \theta = 1 - \sin^2 \theta\). So the expression simplifies to: \(\frac{1 - \sin^2 \theta}{1+\sin \theta}\). This is the final expression in terms of \(\sin \theta\).

Key Concepts

Fundamental Trigonometric Identities

Trigonometric identities are essential in simplifying expressions and solving equations in trigonometry. The fundamental identities are the building blocks for more complex identities. Two critical fundamental identities are:

• Secant (\( \sec \theta \)) is the reciprocal of cosine: \( \sec \theta = \frac{1}{\cos \theta} \).
• Tangent (\( \tan \theta \)) is the ratio of sine to cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

These identities allow us to express trigonometric functions in different forms, enabling simplification. In the original exercise, substituting these into the expression transformed it into a solvable form. This step was crucial, showcasing how applying fundamental identities can break down a complex expression into simpler parts.

Pythagorean Identities

Pythagorean identities are derived from the Pythagorean theorem and play a vital role in trigonometry. They provide a bridge between sine, cosine, and tangent functions. One of the most frequently used Pythagorean identities is:

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

This identity can be rearranged to express other trigonometric functions in terms of sine and cosine, such as:

• \( \cos^2 \theta = 1 - \sin^2 \theta \).

In the original step-by-step solution, this identity was instrumental during simplification. Replacing \( \cos^2 \theta \) with \( 1 - \sin^2 \theta \) helped express the final result entirely in terms of \( \sin \theta \). This exemplifies how powerful Pythagorean identities are for transforming trigonometric expressions.

Trigonometric Simplification

Trigonometric simplification is the process of reducing trigonometric expressions into a simpler form. Simplification often involves using identities to replace certain parts of an expression, making the equations easier to handle. In this context:

• Substitute known identities to change the expression into simpler terms.
• Use arithmetic operations carefully to consolidate terms.
• Rearrange and factor where necessary to achieve the simplest form.

In our example, after applying fundamental identities, we simplified the fraction by performing operations in the denominator and canceling terms. Simplifying trigonometric expressions helps in revealing equivalent forms and solving equations efficiently. It's like peeling layers of an onion to get to the core.

Expressions in Terms of Sine and Cosine

Expressing trigonometric functions in terms of sine and cosine is a common technique to facilitate simplification. Sine and cosine are the basic functions, making other trigonometric expressions more manageable when rewritten in these terms. The key strategies include:

• Using reciprocal identities like \( \sec \theta = \frac{1}{\cos \theta} \).
• Expressing tangent as a ratio of sine to cosine: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

In the given exercise, transforming the expression by substituting for secant and tangent using their definitions allowed a straightforward path to express everything in terms of sine and cosine. This approach streamlined the simplification process further, ultimately yielding the final answer. Understanding how to navigate between trigonometric functions using sine and cosine is fundamental in solving trigonometric problems efficiently.