Problem 54 - Algebra and Trigonometry

Question

Verify each identity. \(\frac{\sin \theta}{1-\cot \theta}-\frac{\cos \theta}{\tan \theta-1}=\sin \theta+\cos \theta\)

Step-by-Step Solution

Answer

The given equation does not hold as an identity.

Step 1: Replace Cotangent and Tangent with Sine and Cosine

Replace the cotangent and tangent in the expression with their equivalent expressions in terms of sine and cosine. Using the identities \(\cot \theta = \frac{1}{\tan \theta}\) and \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) transform the expression to: \(\frac{\sin \theta}{1-\frac{\cos \theta}{\sin \theta}} - \frac{\cos \theta}{\frac{\sin \theta}{\cos \theta} - 1 }\).

Step 2: Simplify each Term

Next, simplify each term by multiplying respective numerator and denominator by reciprocal of the denominator of each term. This simplifies the expression to: \(\sin^2 \theta -\cos^2 \theta\).

Step 3: Evaluate using Pythagorean Identity

Use the identity \(\sin^2 \theta + \cos^2 \theta = 1\) to further simplify the expression. In this case, however, the expression \(\sin^2 \theta -\cos^2 \theta\) can't be directly simplified using this identity.

Step 4: Simplify the Right-Hand Side

The right-hand side of the equation is already in its simplest form, \(\sin \theta + \cos \theta\). It doesn't equate to \(\sin^2 \theta -\cos^2 \theta\), hence the given equation doesn't frame an identity.

Key Concepts

Cotangent

Cotangent, abbreviated as "cot," is a fundamental trigonometric function. It is defined as the reciprocal of the tangent function. In mathematical terms,

• The cotangent of an angle \( \theta \) in a right triangle is the ratio of the adjacent side to the opposite side. This can also be represented using sine and cosine as: \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \).

Understanding cotangent in terms of sine and cosine helps in transforming expressions that involve cotangent. It allows you to leverage these simpler functions to perform transformations and simplify complex trigonometric identities effectively.

Tangent

The tangent function, often simply denoted as "tan," is another primary trigonometric function. Originating from the geometrical concept of a tangent line to a circle,

• Tangent of an angle \( \theta \) in a right triangle is defined as the ratio of the opposite side to the adjacent side. Mathematically, this is \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

The tangent function is particularly pivotal when analyzing periodic properties of trigonometric functions or when transforming trigonometric expressions. Understanding how tangent relates to sine and cosine is crucial when solving problems like verifying trigonometric identities.

Pythagorean Identity

The Pythagorean identity is a fundamental relationship within trigonometry linking the sine and cosine functions. This identity states that:

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

This is immensely useful because it allows any occurrence of \( \sin^2 \theta \) and \( \cos^2 \theta \) in expressions to be simplified or replaced, aiding in the manipulation and simplification of trigonometric expressions. Though it might not directly simplify some equations, like \( \sin^2 \theta - \cos^2 \theta \), knowing related identities can often provide alternative pathways for simplification.

Sine

Sine, abbreviated as "sin," is a fundamental trigonometric function, representing the vertical aspect of an angle in a unit circle.

• The sine of an angle \( \theta \) in a right triangle is the ratio of the length of the opposite side to the hypotenuse. Mathematically expressed as \( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \).

Sine appears frequently in identities and formulas, allowing for expressions involving it and cosine to convert to one another or combine in solving equations. In theoretical and practical applications, understanding sine's role is essential in dissecting trigonometric problems and identities.

Cosine

Cosine, or "cos," is another key trigonometric function often paired with sine. It represents the horizontal component of an angle in the context of a unit circle.

• Cosine of an angle \( \theta \) in a right triangle is the ratio of the adjacent side over the hypotenuse. In formulaic terms, \( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \).

Cosine aids in calculating distances, angles, and is integral to various trigonometric identities and transformations. Understanding cosine and its connection to sine helps in taking apart and solving various trigonometric problems, like verifying the identities or simplifying complex expressions.