Problem 56 - Algebra and Trigonometry

Question

Verify the identity. $$ \cot ^{2} \theta \cos ^{2} \theta=\cot ^{2} \theta-\cos ^{2} \theta $$

Step-by-Step Solution

Answer

The identity is verified: both sides simplify to \( \cos^2 \theta \).

Step 1: Understanding Cotangent and Cosine

Recall that \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Thus, \( \cot^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta} \). This relationship will help us transform the given identity.

Step 2: Substitute Cotangent Identity

Substitute \( \cot^2 \theta \) in the given identity \( \cot^2 \theta \cos^2 \theta = \cot^2 \theta - \cos^2 \theta \) with \( \frac{\cos^2 \theta}{\sin^2 \theta} \). This gives us \( \frac{\cos^4 \theta}{\sin^2 \theta} = \frac{\cos^2 \theta}{\sin^2 \theta} - \cos^2 \theta \).

Step 3: Simplify the Right Side

Factor the expression on the right side: \( \frac{\cos^2 \theta}{\sin^2 \theta} - \cos^2 \theta = \frac{\cos^2 \theta - \cos^4 \theta}{\sin^2 \theta} \).

Step 4: Factor and Simplify Expressions

Notice that \( \cos^2 \theta - \cos^4 \theta = \cos^2 \theta (1 - \cos^2 \theta) = \cos^2 \theta \sin^2 \theta \), because \( 1 - \cos^2 \theta = \sin^2 \theta \) from the Pythagorean identity. Substitute this back, obtaining \( \frac{\cos^2 \theta \sin^2 \theta}{\sin^2 \theta} \).

Step 5: Simplify the Expression

The fraction \( \frac{\cos^2 \theta \sin^2 \theta}{\sin^2 \theta} \) simplifies to \( \cos^2 \theta \), since \( \sin^2 \theta \) in the numerator and denominator cancel each other.

Step 6: Verify Both Sides are Equal

Now, we have \( \frac{\cos^4 \theta}{\sin^2 \theta} = \cos^2 \theta \). Since \( \cos^2 \theta \) was derived from simplifying the original setup, both sides match, confirming the identity.

Key Concepts

Cotangent Identity

The cotangent identity is an essential concept in trigonometry. It relates the cotangent function to the sine and cosine functions. Cotangent, denoted as \( \cot \theta \), is defined as the reciprocal of the tangent. Therefore, it can be expressed as \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

• **Cotangent Square:** When dealing with \( \cot^2 \theta \), simply square both the numerator and the denominator, resulting in \( \cot^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta} \).
• This form is crucial when performing substitutions in trigonometric expressions and equations.

In this exercise, substituting \( \cot^2 \theta \) helps transform the identity into a recognizable form that can be simplified. Understanding how to manipulate this identity is the first step to unraveling complex trigonometric problems, making calculations more straightforward.

Cosine Identity

The cosine identity revolves around understanding the cosine function, \( \cos \theta \), which is one of the primary trigonometric functions related to angles in a right triangle. The cosine of an angle \( \theta \) is the ratio of the adjacent side to the hypotenuse.

• In a squared form, \( \cos^2 \theta \) is often used in trigonometric identities because it connects to other core identities, like the Pythagorean identity.
• Understanding \( \cos^2 \theta \) as part of identities allows us to solve equations involving multiple trigonometric functions.

Recognizing \( \cos^2 \theta \) not just as a number, but as a part of a broader identity, enables simplifications and transformations in trigonometric equations, as seen in this exercise.

Pythagorean Identity

The Pythagorean identity is a cornerstone in trigonometry. It states that for any angle \( \theta \), the relationship \( \sin^2 \theta + \cos^2 \theta = 1 \) holds true. This identity is derived from the Pythagorean theorem in geometry.

• Using this identity, you can express one trigonometric function in terms of another. For instance, \( 1 - \cos^2 \theta = \sin^2 \theta \).
• This transformation is used to simplify expressions and equations.

In the given exercise, using \( 1 - \cos^2 \theta = \sin^2 \theta \) in the manipulation of expressions was crucial. It allowed the simplification of terms and verification of the identity. This shows the power of the Pythagorean identity in transforming and solving trigonometric exercises.

Trigonometric Simplification

Trigonometric simplification involves reducing complex trigonometric expressions or equations to simpler forms using identities and algebraic manipulations.

• In this exercise, the goal was to confirm that both sides of the equation are equal using known trigonometric identities.
• Substitutions, such as replacing \( \cot^2 \theta \) with its known equivalent, and the distribution and cancellation of terms, were essential in achieving simplification.

The key to successful trigonometric simplification is patience and practice. One should identify opportunities for substitution and simplification, like factoring out common terms or canceling excess components of an expression. Successfully simplifying trigonometric expressions solidifies your understanding of the interrelations among trigonometric identities and enhances problem-solving efficiency.