Problem 5 - Algebra 2 and Trigonometry

Question

In \(3-26,\) prove that each equation is an identity. $$ \cot \theta \sin \theta \cos \theta=\cos ^{2} \theta $$

Step-by-Step Solution

Answer

The equation is an identity, as simplifying gives \( \cos^2 \theta = \cos^2 \theta \).

Step 1: Rewrite Given Equation

The problem statement gives us the equation \( \cot \theta \sin \theta \cos \theta = \cos^2 \theta \). Our goal is to prove that this equation is an identity.

Step 2: Express \(\cot\theta\) in Terms of Sine and Cosine

Recall the trigonometric identity that \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Substitute this expression into the given equation: \( \frac{\cos \theta}{\sin \theta} \sin \theta \cos \theta \).

Step 3: Simplify the Left Side

Substitute \( \cot \theta \) into the original equation and simplify: \( \left( \frac{\cos \theta}{\sin \theta} \right) \sin \theta \cos \theta = \cos \theta \cos \theta \). Since \( \sin \theta \) is in both the numerator and denominator, they cancel out, yielding \( \cos^2 \theta \).

Step 4: Confirm the Identity

We are left with \( \cos^2 \theta = \cos^2 \theta \), which is true for all valid values of \( \theta \). Thus, the equation is indeed an identity.

Key Concepts

Cotangent

Cotangent is one of the six fundamental trigonometric functions. In simpler terms, it’s the reciprocal of the tangent function.

• Definition: The cotangent of an angle, written as \( \cot \theta \), is defined as the ratio of the adjacent side to the opposite side in a right triangle.
• In terms of sine and cosine: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

Understanding cotangent can help simplify complex expressions. It’s crucial in expressing trigonometric identities in terms of more familiar functions like sine and cosine. For instance, let's say you have the expression \( \cot \theta \sin \theta \cos \theta \). By substituting \( \cot \theta \) with \( \frac{\cos \theta}{\sin \theta} \), you make the expression easier to work with in solving trigonometric equations. This method becomes a useful tool in mathematical problem-solving.

Sine and Cosine

Sine and cosine are the foundational trigonometric functions, taught at the start of trigonometry courses.

• Sine: The function \( \sin \theta \) measures the ratio of the length of the opposite side to the hypotenuse in a right triangle.
• Cosine: The function \( \cos \theta \) measures the ratio of the length of the adjacent side to the hypotenuse.

Both are periodic functions and have values ranging from -1 to 1. They are not just limited to right triangles—it’s possible to extend their definitions to apply to any angle, using the unit circle approach. In trigonometric identities, expressing complex functions in terms of sine and cosine can simplify calculations and make proving identities much more manageable.

Trigonometric Simplification

Trigonometric simplification involves breaking down complex trigonometric expressions into simpler, more understandable ones. This process often uses fundamental identities like the Pythagorean identities.Consider the expression given in our exercise: \( \cot \theta \sin \theta \cos \theta = \cos^2 \theta \). To simplify:1. Replace \( \cot \theta \) with \( \frac{\cos \theta}{\sin \theta} \).2. You then have: \( \frac{\cos \theta}{\sin \theta} \times \sin \theta \times \cos \theta \).3. The \( \sin \theta \) terms in the numerator and denominator cancel, simplifying to \( \cos \theta \cos \theta \) or \( \cos^2 \theta \).This example shows how transforming an expression into familiar terms can lead to straightforward simplification. Using consistent techniques helps uncover proofs effectively and ensures no steps are skipped. Simplification is a powerful tool, particularly when proving that an equation or identity holds true for all permissible values.