Simplify The Trigonometric Expression (1 + Tanα Tanβ)^2 + (tanα - Tanβ)^2.

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Introduction

In the realm of trigonometry, simplifying complex expressions is a fundamental skill. This article delves into the simplification of the trigonometric expression (1 + tanα tanβ)^2 + (tanα - tanβ)^2, providing a step-by-step guide to arrive at the correct solution. Understanding these simplifications is crucial for various applications in mathematics, physics, and engineering. We will explore the underlying trigonometric identities and algebraic manipulations necessary to transform this expression into a more concise and manageable form. The correct simplification not only provides a clearer understanding of the relationship between the variables but also aids in solving related problems efficiently. This detailed exploration aims to enhance your problem-solving skills and deepen your understanding of trigonometric identities.

Breaking Down the Expression

Our main task is to simplify the given trigonometric expression: (1 + tanα tanβ)^2 + (tanα - tanβ)^2. This expression combines several trigonometric functions and requires a systematic approach to simplify. We will begin by expanding the squares and then applying various trigonometric identities to reduce the expression to its simplest form. The key here is to recognize the relationships between different trigonometric functions and use appropriate identities to eliminate terms and consolidate the expression. The expansion will involve algebraic identities such as (a + b)^2 = a^2 + 2ab + b^2 and (a - b)^2 = a^2 - 2ab + b^2. After expanding, we will look for opportunities to combine like terms and use trigonometric identities such as tan(x) = sin(x)/cos(x) and sec(x) = 1/cos(x) to further simplify the expression. This methodical approach will help us navigate through the complexity and arrive at an elegant solution.

The Importance of Trigonometric Identities

Trigonometric identities are the cornerstone of simplifying trigonometric expressions. These identities provide the necessary tools to manipulate and transform expressions into simpler forms. For example, the Pythagorean identity, sin^2(x) + cos^2(x) = 1, is frequently used to convert between sine and cosine functions. Similarly, identities like sec^2(x) = 1 + tan^2(x) are crucial for simplifying expressions involving tangent and secant functions. In our case, we will primarily use identities related to tangent and secant functions. Understanding and memorizing these identities is essential for any student or professional working with trigonometry. These identities are not just mathematical tools; they also represent fundamental relationships between angles and sides of triangles, which are vital in fields such as navigation, physics, and engineering. By mastering these identities, one can approach complex problems with confidence and solve them efficiently.

Step-by-Step Simplification

To effectively simplify the expression (1 + tanα tanβ)^2 + (tanα - tanβ)^2, we will follow a step-by-step approach, ensuring clarity and precision in each step. This methodical approach not only helps in arriving at the correct solution but also enhances understanding of the underlying principles. Each step will be explained in detail, highlighting the algebraic manipulations and trigonometric identities used.

Step 1: Expanding the Squares

The first step in simplifying the expression is to expand the squared terms. We will use the algebraic identities: (a + b)^2 = a^2 + 2ab + b^2 and (a - b)^2 = a^2 - 2ab + b^2. Applying these identities to our expression, we get:

(1 + tanα tanβ)^2 = 1 + 2 tanα tanβ + tan^2α tan^2β

(tanα - tanβ)^2 = tan^2α - 2 tanα tanβ + tan^2β

This expansion allows us to break down the complex terms into simpler components. The next step will involve combining these expanded forms and looking for opportunities to simplify further. The ability to correctly expand algebraic expressions is crucial in simplifying trigonometric expressions, as it sets the stage for subsequent simplifications using trigonometric identities.

Step 2: Combining the Expanded Terms

Now, we combine the expanded terms from the previous step:

(1 + 2 tanα tanβ + tan^2α tan^2β) + (tan^2α - 2 tanα tanβ + tan^2β)

By combining like terms, we observe that the 2 tanα tanβ and -2 tanα tanβ terms cancel each other out. This simplification is a critical step as it reduces the complexity of the expression significantly. The resulting expression is:

1 + tan^2α tan^2β + tan^2α + tan^2β

This form is much simpler than the original expression and sets us up for the next step, which involves using trigonometric identities to further simplify the expression.

Step 3: Rearranging and Applying Trigonometric Identities

Rearrange the terms to group similar functions together:

1 + tan^2α + tan^2β + tan^2α tan^2β

We can now use the trigonometric identity sec^2(x) = 1 + tan^2(x). Applying this identity to the 1 + tan^2α term, we get:

sec^2α + tan^2β + tan^2α tan^2β

This substitution helps us introduce the secant function, which is crucial for achieving the final simplified form. The next step involves factoring out common terms to further simplify the expression.

Step 4: Factoring and Final Simplification

Observe that we can factor out sec^2α from the expression. To do this, we need to manipulate the expression slightly. We rewrite the expression as:

sec^2α + tan^2β(1 + tan^2α)

Now, we apply the identity sec^2(x) = 1 + tan^2(x) again to the term (1 + tan^2α):

sec^2α + tan2β(sec2α)

Now we can factor out sec^2α:

sec^2α(1 + tan^2β)

Finally, we apply the identity sec^2(x) = 1 + tan^2(x) one last time to the term (1 + tan^2β):

sec2α(sec2β)

Thus, the simplified expression is:

sec^2α sec^2β

This step-by-step simplification demonstrates how algebraic manipulation combined with trigonometric identities can transform a complex expression into a much simpler form. This final form is not only easier to work with but also provides insights into the relationships between the trigonometric functions involved.

Conclusion

In this comprehensive guide, we have successfully simplified the trigonometric expression (1 + tanα tanβ)^2 + (tanα - tanβ)^2 to its simplest form: sec^2α sec^2β. This simplification was achieved through a systematic approach involving the expansion of squares, combining like terms, and applying fundamental trigonometric identities. The key steps included expanding the original expression, identifying and canceling out terms, applying the Pythagorean identity, and factoring out common trigonometric functions. This process not only simplifies the expression but also reinforces the importance of understanding and applying trigonometric identities.

Summary of Key Steps

Let’s briefly summarize the key steps we followed:

  1. Expanding the Squares: We used the algebraic identities (a + b)^2 = a^2 + 2ab + b^2 and (a - b)^2 = a^2 - 2ab + b^2 to expand the squared terms in the original expression.
  2. Combining Like Terms: We combined the expanded terms and canceled out the 2 tanα tanβ and -2 tanα tanβ terms.
  3. Applying Trigonometric Identities: We used the identity sec^2(x) = 1 + tan^2(x) to convert the expression into terms involving secant functions.
  4. Factoring Out Common Terms: We factored out sec^2α to further simplify the expression.
  5. Final Simplification: We applied the identity sec^2(x) = 1 + tan^2(x) one last time to arrive at the final simplified form: sec^2α sec^2β.

These steps highlight the methodical approach required to simplify complex trigonometric expressions. By understanding and applying these steps, one can tackle similar problems with confidence and accuracy.

Importance of Understanding Trigonometric Simplification

The ability to simplify trigonometric expressions is crucial in various fields, including mathematics, physics, and engineering. Simplified expressions are easier to work with and provide clearer insights into the underlying relationships between trigonometric functions. In calculus, simplified trigonometric forms are often necessary for integration and differentiation. In physics, trigonometric functions are used to describe wave phenomena, oscillations, and motion, and simplification is crucial for solving equations and understanding physical systems. In engineering, especially in fields like electrical engineering and signal processing, trigonometric functions are used extensively, and the ability to simplify expressions is essential for analysis and design.

Practice and Further Exploration

To master trigonometric simplification, practice is essential. Work through various examples and try different approaches to simplify expressions. Explore more complex expressions and look for patterns and strategies that can be applied. Additionally, understanding the geometric interpretations of trigonometric functions can provide valuable insights and aid in simplification. For further exploration, consider studying advanced trigonometric identities and their applications. Topics such as the sum and difference formulas, double-angle formulas, and half-angle formulas can further enhance your understanding and problem-solving skills in trigonometry.

In conclusion, the simplification of (1 + tanα tanβ)^2 + (tanα - tanβ)^2 to sec^2α sec^2β is a testament to the power of algebraic manipulation and trigonometric identities. By following a systematic approach and understanding the underlying principles, one can successfully simplify complex expressions and gain a deeper appreciation for the beauty and utility of trigonometry.