Hey guys! Today, we're diving into trigonometry to simplify the expression sec(x) * csc(x) * cos(x). Don't worry, it's not as intimidating as it looks! We'll break it down step-by-step so you can easily follow along. Understanding trigonometric identities is super important in math, physics, and engineering, so let's get started and make sure you grasp this concept.

Understanding the Basic Trigonometric Functions

Before we jump into simplifying the expression, let's refresh our understanding of the basic trigonometric functions. These functions form the building blocks of trigonometry, and knowing them well will make simplifying expressions much easier. There are six primary trigonometric functions: sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). Each of these functions relates the angles of a right triangle to the ratios of its sides.

• Sine (sin): The sine of an angle in a right triangle is the ratio of the length of the opposite side to the length of the hypotenuse. In other words, sin(x) = Opposite / Hypotenuse.
• Cosine (cos): The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. So, cos(x) = Adjacent / Hypotenuse.
• Tangent (tan): The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. Thus, tan(x) = Opposite / Adjacent. Also, it's important to remember that tan(x) can also be expressed as sin(x) / cos(x).
• Cosecant (csc): The cosecant is the reciprocal of the sine function. This means csc(x) = 1 / sin(x) = Hypotenuse / Opposite.
• Secant (sec): The secant is the reciprocal of the cosine function. Therefore, sec(x) = 1 / cos(x) = Hypotenuse / Adjacent.
• Cotangent (cot): The cotangent is the reciprocal of the tangent function. So, cot(x) = 1 / tan(x) = Adjacent / Opposite. Also, cot(x) can be expressed as cos(x) / sin(x).

Understanding these definitions and relationships is crucial for simplifying trigonometric expressions. Make sure you have a good grasp of these before moving on, as they will be used extensively in the simplification process. By knowing the reciprocals and ratios, you'll be able to manipulate and simplify complex expressions with ease. For example, recognizing that sec(x) is 1/cos(x) allows you to replace sec(x) in any expression, potentially leading to significant simplification. Similarly, understanding that csc(x) is 1/sin(x) and tan(x) is sin(x)/cos(x) will enable you to rewrite expressions in different forms, making them easier to work with. Remember, trigonometry is all about these relationships, so mastering them is key to success!

Rewriting the Expression Using Basic Definitions

Now that we've recapped the basic trigonometric functions, let's rewrite the expression sec(x) * csc(x) * cos(x) using these definitions. This step is crucial because it allows us to express the entire expression in terms of sine and cosine, which are often easier to manipulate. By converting sec(x) and csc(x) into their reciprocal forms, we can simplify the expression more effectively. Remember, sec(x) is the reciprocal of cos(x), and csc(x) is the reciprocal of sin(x). So we can rewrite the expression as follows:

sec(x) * csc(x) * cos(x) = (1 / cos(x)) * (1 / sin(x)) * cos(x)

This transformation is the heart of our simplification strategy. By expressing everything in terms of sine and cosine, we open the door to potential cancellations and further simplifications. Notice how the original expression involves three different trigonometric functions, but after rewriting, we only have sine and cosine to deal with. This makes the expression more manageable and easier to simplify. For instance, if we had an expression involving tan(x), rewriting it as sin(x) / cos(x) would similarly allow us to work with just sine and cosine. The goal here is to reduce the complexity of the expression by using these basic definitions. Once you get comfortable with these transformations, you'll be able to quickly simplify a wide variety of trigonometric expressions. This step is not just about rewriting; it's about making the expression more accessible and easier to understand, setting the stage for the final simplification.

Simplifying the Expression

Now that we have rewritten the expression as (1 / cos(x)) * (1 / sin(x)) * cos(x), we can proceed with simplifying it. This involves looking for opportunities to cancel out terms and reduce the expression to its simplest form. In this case, we have a cos(x) in the numerator and a cos(x) in the denominator, which allows us to cancel them out. This cancellation is a key step in simplifying the expression.

So, let's perform the cancellation:

(1 / cos(x)) * (1 / sin(x)) * cos(x) = (1 / sin(x)) * (cos(x) / cos(x)) = (1 / sin(x)) * 1 = 1 / sin(x)

After canceling out the cos(x) terms, we're left with 1 / sin(x). Now, remember the definition of the cosecant function? csc(x) is defined as 1 / sin(x). Therefore, we can further simplify the expression:

1 / sin(x) = csc(x)

So, the simplified form of sec(x) * csc(x) * cos(x) is csc(x). This final simplification highlights the power of using trigonometric identities. By rewriting the original expression in terms of sine and cosine and then canceling out common terms, we were able to reduce it to a single trigonometric function. This not only makes the expression easier to understand but also more manageable in further calculations. For example, if you were integrating this expression, integrating csc(x) would be much simpler than integrating the original expression. Mastering these simplification techniques is invaluable for solving more complex problems in trigonometry and calculus. Remember to always look for opportunities to rewrite expressions using basic definitions and to cancel out common terms to arrive at the simplest form.

Final Answer

Therefore, after simplifying the expression sec(x) * csc(x) * cos(x), we arrive at the final answer:

sec(x) * csc(x) * cos(x) = csc(x)

In summary, the key steps were:

1. Rewrite sec(x) as 1 / cos(x) and csc(x) as 1 / sin(x).
2. Cancel out the cos(x) terms.
3. Recognize that 1 / sin(x) is equal to csc(x).

Understanding trigonometric identities and being able to manipulate them is crucial for simplifying complex expressions. By following these steps, you can easily simplify similar expressions and gain a deeper understanding of trigonometry. Remember to practice these techniques to become more proficient. The more you work with these identities, the easier it will become to recognize opportunities for simplification. Keep practicing, and you'll master these concepts in no time! This simplification not only provides a more concise form of the expression but also enhances your understanding of the relationships between different trigonometric functions. By mastering these techniques, you'll be well-equipped to tackle more advanced problems in mathematics, physics, and engineering. Keep up the great work!