Problem 27 - Algebra and Trigonometry

Question

Verify the identity. $$ \frac{\cos u \sec u}{\tan u}=\cot u $$

Step-by-Step Solution

Answer

Yes, the identity is verified.

Step 1: Understanding the problem

The given identity to verify is \( \frac{\cos u \sec u}{\tan u}=\cot u \). To verify an identity, we need to show that the left side of the equation can be simplified to look like the right side.

Step 2: Simplify using identities

Remember the trigonometric identities: \( \sec u = \frac{1}{\cos u} \) and \( \tan u = \frac{\sin u}{\cos u} \). We will use these to simplify the left hand side.

Step 3: Substituting identities

Replace \( \sec u \) with \( \frac{1}{\cos u} \) in the expression. The expression becomes \( \frac{\cos u \cdot \frac{1}{\cos u}}{\tan u} \). Simplifying, this is \( \frac{1}{\tan u} \).

Step 4: Substitute \( \tan u \) identity

Substitute \( \tan u = \frac{\sin u}{\cos u} \) into the expression. So, \( \frac{1}{\tan u} = \frac{1}{\frac{\sin u}{\cos u}} \). Simplifying, this becomes \( \frac{\cos u}{\sin u} \).

Step 5: Simplify to \( \cot u \)

Recognize that \( \frac{\cos u}{\sin u} = \cot u \). Thus, we have shown that the left side simplifies to exactly the right side, \( \cot u \), verifying the identity.

Key Concepts

Trigonometric Functions

Trigonometric functions form the building blocks of trigonometry. These functions include sine, cosine, tangent, cotangent, secant, and cosecant. They are defined as ratios of the sides of a right triangle or as coordinates of a point on the unit circle.

In this exercise, we encounter the cosine (\(\cos u\)), tangent (\(\tan u\)), secant (\(\sec u\)), and cotangent (\(\cot u\)) functions. Important relationships include:

• Cosine function: Approximately measures the adjacent side of a right triangle divided by the hypotenuse.
• Secant: Reciprocal of cosine, \(\sec u = \frac{1}{\cos u}\).
• Tangent: Ratio of sine to cosine, \(\tan u = \frac{\sin u}{\cos u}\).
• Cotangent: Reciprocal of tangent, which is \(\cot u = \frac{1}{\tan u} = \frac{\cos u}{\sin u}\).

Understanding these relationships helps simplify expressions effectively.

Simplifying Expressions

Simplifying trigonometric expressions involves transforming them to equivalent and often simpler forms using known identities and relationships. The main goal is to reduce the expression by removing complex fractions or substituting equivalent ratios.

In this exercise, you start by substituting identities for secant and tangent:

• Replace \(\sec u\) with \(\frac{1}{\cos u}\).
• Replace \(\tan u\) with \(\frac{\sin u}{\cos u}\).

This substitution allows the expression on the left side to be written in terms of cosine and sine, making it easier to simplify further.

Using these substitutions helps reduce the expression \(\frac{\cos u \cdot \frac{1}{\cos u}}{\tan u}\) to \(\frac{1}{\tan u}\), and then to \(\cot u\) after additional simplification.

Verifying Identities

Verifying trigonometric identities means proving one side of an equation is equal to the other. It often involves transforming one side of the equation using identities until it matches the other.

This process can include:

• Substituting equivalent expressions.
• Simplifying complex fractions.
• Rewriting terms to reveal hidden identities.

In our exercise, we started with the identity \(\frac{\cos u \sec u}{\tan u} = \cot u\). The solution showed how to manipulate the left side by substituting known identities until it resolved to \(\cot u\), thus verifying the given identity correctly. This method confirms the equality by logical steps and transformations.

Trigonometric Ratios

Trigonometric ratios relate the angles and sides of right triangles. They are foundational definitions in trigonometry and essential for solving various mathematical problems. Key trigonometric ratios include sine, cosine, tangent, and their reciprocals (cosecant, secant, and cotangent).

In this exercise, the focus is on these ratios:

• Tangent as the ratio of opposite to adjacent side or sine to cosine.
• Cotangent, being the reciprocal of tangent or the ratio of adjacent to opposite side.

Understanding these allows us to transform one function into another and is crucial for simplifying and verifying identities. Familiarity with these ratios and their properties enables us to tackle trigonometric expressions with confidence, as seen in our exercise verification.