Problem 55 - Precalculus with Calculus Previews

Question

Verify the given identity. $$ (1-\tan \beta)^{2}(1+\tan \beta)^{2}+4 \tan ^{2} \beta=\sec ^{4} \beta $$

Step-by-Step Solution

Answer

The identity is verified as both sides simplify to \(\sec^4 \beta\).

Step 1: Expand the Expression

Start by expanding the left side of the equation \[(1-\tan \beta)^2(1+\tan \beta)^2 + 4 \tan^2 \beta.\]This expression can be seen as \(((1-\tan \beta)(1+\tan \beta))^2\), which follows the difference of squares identity. Therefore, it simplifies to \((1 - \tan^2 \beta)^2\).

Step 2: Use Trigonometric Identity

Recall the trigonometric identity for tangent and secant: \[\tan^2 \beta = \sec^2 \beta - 1.\] Use this identity to replace \(\tan^2 \beta\) in the expanded expression. This makes the expression:\[((1 - (\sec^2 \beta - 1)))^2 + 4(\sec^2 \beta - 1).\] This simplifies to:\[(2 - \sec^2 \beta)^2 + 4 (\sec^2 \beta - 1).\]

Step 3: Simplify the Expression

Simplify the expression further. Expanding \((2 - \sec^2 \beta)^2\) gives:\[4 - 4\sec^2 \beta + \sec^4 \beta.\] Adding \(4(\sec^2 \beta - 1)\) to this:\[4 - 4\sec^2 \beta + \sec^4 \beta + 4\sec^2 \beta - 4.\] This simplifies to \(\sec^4 \beta\) after canceling and combining terms.

Key Concepts

Difference of Squares

The difference of squares is a crucial concept in algebra and trigonometry. It's a formula that helps to simplify expressions and solve equations. The formula is:

• (a - b)(a + b) = a^2 - b^2

In this exercise, we apply it to the expression \((1 - \tan \beta)(1 + \tan \beta)\). When simplified, it becomes \(1 - \tan^2 \beta\). This step changes a complex multiplication into a simpler subtraction, making it easier to handle in further calculations.
By turning a product into a simpler form, you're using the power of the difference of squares for effective problem-solving. This transformation underpins many algebraic and trigonometric simplifications, enabling us to move from complex to more manageable expressions.

Tangent and Secant Relationship

The relationship between tangent and secant is important in trigonometry. It's represented by the identity:

• \(\tan^2 \beta = \sec^2 \beta - 1\)

This identity is derived from the Pythagorean identity for sine and cosine: \(\sin^2 \beta + \cos^2 \beta = 1\). When dealing with tangent and secant (which are just sine over cosine and the inverse of cosine, respectively), transformations of this nature are crucial.
In this problem, we use the identity to replace \(\tan^2 \beta\) with \(\sec^2 \beta - 1\) in the expression. It allows us to simplify and solve trigonometric equations more efficiently by involving only one trigonometric function. This step is fundamental in reducing the original equation into something that is in terms of secant squared, making comparison straightforward.

Expression Simplification

Simplifying complex expressions is essential in mathematics for clarity and to facilitate problem solving. In our problem, after expanding and substituting the terms, the expression becomes:

• \((2 - \sec^2 \beta)^2 + 4(\sec^2 \beta - 1)\)

The process involves expanding these expressions to manage and combine like terms effectively. By expanding \((2 - \sec^2 \beta)^2\), we get:

• \(4 - 4\sec^2 \beta + \sec^4 \beta\)

Adding the second part \(4(\sec^2 \beta - 1)\), which simplifies to \(4\sec^2 \beta - 4\), allows all terms to be combined. The simplification process is powerful because it reduces complexity, often revealing symmetric or structural properties of the expressions that were not obvious initially, such as canceling terms resulting in the final form \(\sec^4 \beta\).

Trigonometric Equations

Trigonometric equations often require the application of identities and simplification strategies to solve. The goal is to verify or identify equalities involving trigonometric expressions. In this exercise, after simplification, you reach the identity:

• \(\sec^4 \beta = \sec^4 \beta\)

This confirms that the initial equation holds true.
Solving such equations involves:

• Recognizing identities and using them to transform expressions.
• Combining like terms.
• Simplifying from complex to simpler forms using algebraic techniques.

These practices are essential in verifying identities and solving function-based equations in trigonometry. Understanding these methods helps you to tackle a wide range of related mathematical problems with confidence.