Problem 11 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-50: 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: Express Trigonometric Functions in Terms of Sine and Cosine

To verify the identity \((\sec u - \tan u)(\csc u + 1) = \cot u\), start by expressing \(\sec u\), \(\tan u\), and \(\csc u\) 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 expression:\(\left(\frac{1}{\cos u} - \frac{\sin u}{\cos u}\right)\left(\frac{1}{\sin u} + 1\right)\)

Step 2: Simplify the Expression

Simplify the expression by combining terms:Inside the first parentheses: \(\frac{1}{\cos u} - \frac{\sin u}{\cos u} = \frac{1 - \sin u}{\cos u}\).Inside the second parentheses: \(\frac{1}{\sin u} + 1 = \frac{1 + \sin u}{\sin u}\).The expression becomes:\(\frac{1 - \sin u}{\cos u} \cdot \frac{1 + \sin u}{\sin u}\)

Step 3: Apply the Difference of Squares

Notice that the expression \((1 - \sin u)(1 + \sin u)\) resembles the difference of squares, which simplifies to \(1 - \sin^2 u\). Since \(1 - \sin^2 u = \cos^2 u\), the expression now becomes:\(\frac{\cos^2 u}{\cos u \sin u}\)

Step 4: Simplify to Obtain the Cotangent

Now, simplify the expression \(\frac{\cos^2 u}{\cos u \sin u}\). Reduce by cancelling one \(\cos u\) from the numerator and the denominator to get:\(\frac{\cos u}{\sin u}\)This is the definition of \(\cot u\), because \(\cot u = \frac{\cos u}{\sin u}\). Thus, the original identity \((\sec u - \tan u)(\csc u + 1) = \cot u\) is verified.

Key Concepts

sine and cosine

In trigonometry, sine and cosine are fundamental functions that relate the angles of a triangle to the lengths of its sides in the unit circle. These functions are part of the building blocks for various other trigonometric identities and expressions.

Understanding their relationship:

• The sine function, \( ext{sin } u\), gives the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.
• The cosine function, \( ext{cos } u\), gives the ratio of the adjacent side to the hypotenuse.

These relationships form the basis of the other identity transformations. For instance:

• \( ext{sec } u = \frac{1}{\cos u}\)
• \( ext{tan } u = \frac{\sin u}{\cos u}\)
• \( ext{csc } u = \frac{1}{\sin u}\)

Using sine and cosine to transform trigonometric expressions can greatly simplify equations, making them easier to manipulate and solve.

difference of squares

The difference of squares is a valuable technique in algebra and trigonometry, used to simplify expressions and solve equations. It refers to a specific type of multiplication formula:

Recall the formula: \((a - b)(a + b) = a^2 - b^2\)

This algebraic identity can be conveniently applied in trigonometric simplifications, especially when you see structures like \( (1 - \sin u)(1 + \sin u) \).

In our context:

• Applying the difference of squares to \( (1 - \sin u)(1 + \sin u) \) yields \( 1 - \sin^2 u \).
• Using the Pythagorean identity, \( 1 - \sin^2 u = \cos^2 u \).

This allows the expression to transform and simplify into terms that are easier to manage, reflecting the potent link between algebra and trigonometry.

cotangent

Cotangent is one of the six primary trigonometric functions, explained as the ratio of the adjacent side to the opposite side in a right triangle. Mathematically, it is defined as:

\(\cot u = \frac{\cos u}{\sin u}\)

This definition reveals that cotangent is essentially the reciprocal of the tangent function, expressed as:

• \(\cot u = \frac{1}{\tan u}\)

The role of cotangent is critical in simplifying trigonometric expressions. By converting expressions in terms of cotangent, complicated equations can often be handled more effectively.

In the original exercise, simplifying the expression to match cotangent helps achieve the final verification of the identity:

• The resulting expression, \( \frac{\cos u}{\sin u} \), directly corresponds to \( \cot u \).

This confirms our understanding and use of trigonometric identities to simplify complex expressions.

trigonometric expression simplification

Trigonometric expressions often appear complex due to the combination of various functions, angles, and identities. The goal of simplification is to reduce these expressions to their simplest form, often to verify equalities or make calculations manageable.

Key techniques for simplifying trigonometric expressions involve:

• Expressing complex functions in terms of sine and cosine for uniformity.
• Applying algebraic identities, like the difference of squares, to simplify terms.
• Cancelling common factors in numerators and denominators where possible.

In our exercise, the use of these strategies was crucial in verifying the identity:

By expressing each function in terms of sine and cosine and applying the difference of squares, we efficiently reduced the expression to obtain cotangent.

Simplification not only aids in verifying identities but also in solving trigonometric equations, making these methods invaluable across a wide range of mathematical problems.