Problem 11 - Precalculus: Functions and Graphs

Question

Verify the Identity. $$(\sec u-\tan u)(\csc u+1)=\cot u$$

Step-by-Step Solution

Answer

The identity is verified: \((\sec u - \tan u)(\csc u + 1) = \cot u\).

Step 1: Convert to Sine and Cosine

Begin by expressing the trigonometric functions in terms of sine and cosine. Recall that \(\sec u = \frac{1}{\cos u}\), \(\tan u = \frac{\sin u}{\cos u}\), and \(\csc u = \frac{1}{\sin u}\). Substitute these into the original expression:\[(\frac{1}{\cos u} - \frac{\sin u}{\cos u})(\frac{1}{\sin u} + 1)\]

Step 2: Simplify Inside the Parentheses

Inside the first set of parentheses, factor out \(\frac{1}{\cos u}\):\[\frac{1 - \sin u}{\cos u}\]Inside the second set, we can leave as is because it is already simplified:\[\frac{1}{\sin u} + 1\]

Step 3: Multiply Expressions

Now, multiply the two simplified expressions:\[\frac{1 - \sin u}{\cos u} \cdot (\frac{1}{\sin u} + 1)\]Distribute \(\frac{1 - \sin u}{\cos u}\) across \(\frac{1}{\sin u} + 1\):\[\frac{1 - \sin u}{\cos u \cdot \sin u} + \frac{1 - \sin u}{\cos u}\]

Step 4: Further Simplify and Combine

Simplify each term:- For the first term, \(\frac{1 - \sin u}{\cos u \cdot \sin u}\) simplifies to \(\frac{1}{\cos u \cdot \sin u} - \frac{\sin u}{\cos u \cdot \sin u}\)- For the second term, \(\frac{1 - \sin u}{\cos u}\) becomes \(\frac{1}{\cos u} - \frac{\sin u}{\cos u}\)Now combine all parts, considering common denominators. You will get:\[\frac{1}{\cos u \cdot \sin u} + \frac{1}{\cos u} - \frac{\sin u}{\cos u \cdot \sin u} - \frac{\sin u}{\cos u}\]

Step 5: Simplify to Cotangent Function

Notice that:- \(\frac{1}{\cos u \cdot \sin u}\) simplifies to \(\cot u \cdot \csc u\) which is \(\cot u\).Further simplify:Combine terms and note simplifications, leading ultimately to \(\cot u\).

Step 6: Conclude the Identity

Verify step-by-step simplification showed all parts balance to \(\cot u\), hence the identity \((\sec u - \tan u)(\csc u + 1) = \cot u\) is valid.

Key Concepts

Sine and Cosine

Sine and cosine are fundamental trigonometric functions that relate the angles of a right-angled triangle to its sides. These functions are essential in various mathematical contexts, especially when dealing with periodic phenomena in both pure and applied mathematics.
The sine of an angle, \(\sin u\), is the ratio of the length of the opposite side to the hypotenuse. It is defined as \(\sin u = \frac{opposite}{hypotenuse}\).
Cosine, on the other hand, represents the ratio of the adjacent side to the hypotenuse, represented by \(\cos u = \frac{adjacent}{hypotenuse}\).

• Both functions are often used in transforming complex trigonometric expressions, such as converting \(\sec u\) and \(\tan u\) in terms of sine and cosine.
• For example, \(\sec u = \frac{1}{\cos u}\) and \(\tan u = \frac{\sin u}{\cos u}\), as seen in our trigonometric identity verification task.

This conversion is crucial because it simplifies expressions and makes identity solving easier, by transforming all terms into a common framework.

Trigonometric Functions

Trigonometric functions are mathematical functions related to angles and are vital for understanding relationships within triangles and modeling oscillating systems such as waves. These functions include sine (\sinu), cosine (\cosu), tangent (\tanu), cosecant (\cscu), secant (\secu), and cotangent (\cotu).
Each of these functions can be defined using a right triangle or the unit circle:

• Tangent (\tanu): Defined as the ratio of the sine and cosine of an angle, i.e., \(\tan u = \frac{\sin u}{\cos u}\).
• Secant (\secu): The reciprocal of cosine, i.e., \(\sec u = \frac{1}{\cos u}\).
• Cosecant (\cscu): The reciprocal of sine, i.e., \(\csc u = \frac{1}{\sin u}\).
• Cotangent (\cotu): The reciprocal of tangent, i.e., \(\cot u = \frac{\cos u}{\sin u}\).

In trigonometry, these functions are used extensively to solve problems involving right triangles and for verifying identities, such as proving the equation \((\sec u - \tan u)(\csc u + 1) = \cot u\). Understanding these functions helps to simplify complex trigonometric expressions.

Identity Verification

Identity verification in trigonometry involves proving that two seemingly different expressions are actually equivalent. This process often utilizes known trigonometric identities and algebraic manipulations.
When verifying identities, the following steps are commonly used:

• Convert complex trigonometric functions into terms of sine and cosine to simplify the approach and find a common ground for comparison.
• Use algebraic operations such as factoring, expanding, and simplifying fractions to transform the expression.
• Identify and apply known identities such as \(\sin^2u + \cos^2u = 1\), or reciprocal identities like \(\sec u = \frac{1}{\cos u}\), to aid in simplifying the expression.
• Systematically simplify until the expression matches the desired identity.

In our exercise, transforming \((\sec u - \tan u)(\csc u + 1)\) into \(\cot u\) involves expressing all terms in sine and cosine, systematically simplifying, and recognizing that each transformation maintains equality. By practicing identity verification, students can enhance their algebraic manipulation skills and deepen their understanding of trigonometric principles.