Problem 154 - Calculus Volume 1

Question

Verify that each equation is an identity. \(\quad \frac{\tan \theta-\cot \theta}{\sin \theta \cos \theta}=\sec ^{2} \theta-\csc ^{2} \theta\)

Step-by-Step Solution

Answer

The identity is verified.

Step 1: Rewrite Trigonometric Functions

Start by rewriting the terms \( \tan \theta \) and \( \cot \theta \) in the left-hand side of the equation using their fundamental identities. \[ \tan \theta = \frac{\sin \theta}{\cos \theta} \quad \text{and} \quad \cot \theta = \frac{\cos \theta}{\sin \theta} \]Substituting these identities gives:\[ \frac{\frac{\sin \theta}{\cos \theta} - \frac{\cos \theta}{\sin \theta}}{\sin \theta \cos \theta} \]

Step 2: Combine the Fractions

Combine the fractions in the numerator by finding a common denominator. The common denominator for the fractions is \( \sin \theta \cos \theta \):\[ \frac{\sin^2 \theta - \cos^2 \theta}{\cos \theta \sin \theta} \]

Step 3: Simplify the Fraction

The numerator itself can now be simplified to:\[ \sin^2 \theta - \cos^2 \theta \]Divide each component by the \(\sin \theta \cos \theta\):\[ \frac{\sin^2 \theta - \cos^2 \theta}{\sin \theta \cos \theta} \]

Step 4: Separate the Fractions

Split the single fraction into two separate fractions:\[ \frac{\sin^2 \theta}{\sin \theta \cos \theta} - \frac{\cos^2 \theta}{\sin \theta \cos \theta} \]Simplifying gives:\[ \frac{\sin \theta}{\cos \theta} - \frac{\cos \theta}{\sin \theta} \]

Step 5: Simplify Using Pythagorean Identity

Recall the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \) and relate it to the expression:The simplified form of our fractions becomes:\[ \sec^2 \theta - \csc^2 \theta\] Now observe that this expression matches the right-hand side of the original identity, proving it to be true.

Key Concepts

Fundamental Trigonometric Functions

Trigonometric functions are fundamental in mathematics due to their extensive applications in trigonometry. The three primary functions are sine (\( \sin \theta \)), cosine (\( \cos \theta \)), and tangent (\( \tan \theta \)). These functions relate angles in a right triangle to the ratios of its sides.

• Sine (\( \sin \theta \)): Ratio of the length of the opposite side to the hypotenuse of the triangle.
• Cosine (\( \cos \theta \)): Ratio of the length of the adjacent side to the hypotenuse of the triangle.
• Tangent (\( \tan \theta \)): Ratio of the length of the opposite side to the adjacent side.

Other functions like cotangent (\( \cot \theta \)), secant (\( \sec \theta \)), and cosecant (\( \csc \theta \)) are derived from these primary functions by taking their reciprocals.
These relationships form the backbone of trigonometric identities and equations, allowing for complex expressions to be manipulated and simplified.

Trigonometric Equations

Trigonometric equations involve trigonometric functions and are solved to find the angle values that satisfy the equation. Understanding these equations is crucial because it allows you to find the angles for certain conditions directly.
To solve a trigonometric equation, one typically needs to:

• Rewrite each term using known identities, which can simplify the problem greatly.
• Combine like terms or use algebraic techniques to solve for the variable, usually the angle.
• Check the solutions in the original equation to ensure they are valid.

Trigonometric equations often have multiple solutions because trigonometric functions are periodic. This periodic nature means the solution can recur at regular intervals, leading to potentially infinite solutions depending on the domain of interest.

Pythagorean Identity

One of the most well-known trigonometric identities is the Pythagorean Identity, which is rooted in the Pythagorean Theorem.
It states that for any angle \( \theta \), the following identity holds:\[ \sin^2 \theta + \cos^2 \theta = 1 \]This identity is crucial because it relates sine and cosine, making it easier to express one in terms of the other.
Additionally, two other Pythagorean identities derived from the primary one are:

• \( 1 + \tan^2 \theta = \sec^2 \theta \)
• \( 1 + \cot^2 \theta = \csc^2 \theta \)

These identities are essential tools in verifying equations, simplifying expressions, and solving trigonometric equations, as demonstrated in the original exercise.

Trigonometric Simplification

Simplification in trigonometry involves reducing complex expressions to simpler or more easily interpretable forms. This often requires using known identities.
To simplify a trigonometric expression effectively:

• Substitute basic identities where possible.
• Use algebraic manipulations such as finding common denominators and factoring.
• Apply Pythagorean identities to relate sine to cosine or other similar functions.
• Convert functions to sine and cosine when it helps to streamline the process.

In our original step-by-step solution, simplification was key in proving the trigonometric identity. By rewriting terms and splitting complex fractions, the solution matched both sides of the equation. Simplification not only clarifies expressions but also helps reveal hidden equivalencies within trigonometric functions.