Problem 54 - 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 as both sides simplify to \( \frac{\cos^4 \theta}{\sin^2 \theta} \).

Step 1: Start with the Left-Hand Side (LHS)

First, look at the LHS of the equation, which is \[ \cot ^{2} \theta \cos ^{2} \theta. \] Let's start simplifying this expression.

Step 2: Substitute Trigonometric Identity

Recall that \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Therefore, \( \cot^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta} \). Substitute this into the LHS:\[ \cot^2 \theta \cos^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta} \cdot \cos^2 \theta = \frac{\cos^4 \theta}{\sin^2 \theta}. \]

Step 3: Simplify the Expression

Now the expression becomes \( \frac{\cos^4 \theta}{\sin^2 \theta} \). Notice the trigonometric identity \( \sin^2 \theta = 1 - \cos^2 \theta \), replace \( \sin^2 \theta \):\[ \frac{\cos^4 \theta}{1 - \cos^2 \theta}. \]

Step 4: Start Simplifying Right-Hand Side (RHS)

The RHS of the original equation is \( \cot^2 \theta - \cos^2 \theta \). Substitute the identity for \( \cot^2 \theta \), which is \( \frac{\cos^2 \theta}{\sin^2 \theta} \), on the RHS:\[ \frac{\cos^2 \theta}{\sin^2 \theta} - \cos^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta} - \frac{\cos^2 \theta \cdot \sin^2 \theta}{\sin^2 \theta}. \]

Step 5: Simplify Expression Using a Common Denominator

Combine the expression under a common denominator:\[ \frac{\cos^2 \theta - \cos^2 \theta \cdot \sin^2 \theta}{\sin^2 \theta}. \] Factor out \( \cos^2 \theta \) in the numerator:\[ \frac{\cos^2 \theta (1 - \sin^2 \theta)}{\sin^2 \theta}. \] Use the identity \( 1 - \sin^2 \theta = \cos^2 \theta \):\[ \frac{\cos^2 \theta \cdot \cos^2 \theta}{\sin^2 \theta} = \frac{\cos^4 \theta}{\sin^2 \theta}. \]

Step 6: Verify LHS = RHS

Now that both the LHS and RHS simplify to \( \frac{\cos^4 \theta}{\sin^2 \theta} \), we have verified:\[ \cot ^{2} \theta \cos ^{2} \theta = \frac{\cos^4 \theta}{\sin^2 \theta} = \cot ^{2} \theta - \cos ^{2} \theta \] Therefore, the identity is verified.

Key Concepts

Cotangent

The cotangent is a fundamental trigonometric function that you will encounter often. It's abbreviated as "cot" and is related to the basic sine and cosine functions. Cotangent is defined as the ratio of the cosine of an angle \( \theta \) to the sine of that angle. In formula terms, this is expressed as:

• \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

This means whenever you see cotangent, you can think of it as comparing the horizontal component (cosine) to the vertical component (sine) of an angle on the unit circle.
An important thing to note is that cotangent is undefined whenever \( \theta \) is such that \( \sin \theta = 0\). In the unit circle, these are angles where the sine value is zero, such as \( \theta = 0, \pi, 2\pi, \) etc.
In terms of simplification, expressing cotangent in terms of sine and cosine, as we did with the identity \( \cot^2 \theta = \frac{\cos^2 \theta}{\sin^2 \theta} \), helps in simplifying more complex trigonometric expressions.

Cosine

Cosine, abbreviated as "cos," is one of the primary trigonometric functions that measure the horizontal component of an angle on the unit circle.

• \( \cos \theta \) represents the x-coordinate of a point on the unit circle for a given angle \( \theta \).

The cosine function shows how a circle rotates, describing an oscillation over a cycle of \( 2\pi \). It starts at 1 when \( \theta = 0\) and decreases to -1 at \( \theta = \pi\), back to 1 at \( \theta = 2\pi\).
It is crucial for solving identities, as cosine's squared value \( \cos^2 \theta \) is often used in transformation formulas, like the Pythagorean identity:

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

This relationship helps when simplifying expressions, as it allows one to replace \( \sin^2 \theta \) or \( \cos^2 \theta \) in terms of the other. For example, in our exercise, we used \( 1 - \cos^2 \theta = \sin^2 \theta \) to replace \( \sin^2 \theta \) and simplify our equation.

Trigonometric Simplification

Trigonometric simplification is a process of reducing complex trigonometric expressions using known identities and algebraic manipulations. It involves recognizing patterns and applying identities such as Pythagorean, reciprocal, and cofunction identities to simplify expressions.
Consider the identity we worked on: \( \cot^2 \theta \cos^2 \theta = \cot^2 \theta - \cos^2 \theta \). The goal was to show that both sides of the equation were equal by breaking them down into simpler forms.
Here's a brief breakdown of how simplification happened:

• We expressed \( \cot^2 \theta \) using the \( \cos^2 \theta \) and \( \sin^2 \theta \).
• Replaced the term \( \sin^2 \theta \) using the identity \( 1 - \cos^2 \theta = \sin^2 \theta\).
• Brought terms to a common denominator to allow easy factoring and combining.
• Finally, demonstrated that both original and simplified expressions equate to form the same outcome, verifying the identity.

Simplifying trigonometric identities requires practice in applying these identities correctly. Developing this skill assists in solving a wide range of mathematical problems more efficiently.