Simplifying The Trigonometric Expression 1 - Sin(x) * Cos(x) * Tan(x) A Step-by-Step Guide

October 10, 2025 by Scholario Team 91 views

Hey guys! Have you ever stumbled upon a trigonometric expression that looks like a jumbled mess? Well, you're not alone! Trigonometric expressions can seem daunting at first, but with a few key concepts and a systematic approach, you can simplify them like a pro. In this article, we're going to break down the simplification of the expression 1 - sin(x) * cos(x) * tan(x). We'll walk through each step, explaining the reasoning behind it, so you can confidently tackle similar problems in the future. Get ready to dive into the world of trigonometry and unlock the secrets of simplification!

Understanding the Basics: Trigonometric Identities

Before we jump into the simplification process, let's refresh our understanding of some fundamental trigonometric identities. These identities are the building blocks of trigonometric manipulation, and knowing them is crucial for simplifying complex expressions. Think of them as the essential tools in your trigonometric toolbox. You'll need these key identities to make simplifying easier.

The Tangent Identity: This is perhaps the most important identity for this particular problem. It states that tan(x) is equal to sin(x) divided by cos(x). Mathematically, we write this as: tan(x) = sin(x) / cos(x). This identity allows us to express the tangent function in terms of sine and cosine, which is often the first step in simplifying expressions involving tangent.
The Pythagorean Identity: This identity is a cornerstone of trigonometry and relates sine and cosine. It states that the square of the sine of an angle plus the square of the cosine of the same angle equals 1. In mathematical terms: sin²(x) + cos²(x) = 1. This identity is incredibly versatile and can be rearranged to express either sin²(x) or cos²(x) in terms of the other, making it invaluable for simplification.

Understanding these identities is like having a cheat sheet for trigonometry. They allow you to rewrite expressions in different forms, making it easier to spot opportunities for simplification. Keep these identities in mind as we move on to the next step!

Step 1: Expressing tan(x) in Terms of sin(x) and cos(x)

The first step in simplifying the expression 1 - sin(x) * cos(x) * tan(x) is to use the tangent identity we just discussed. Remember, tan(x) = sin(x) / cos(x). By substituting this into our expression, we replace the tangent function with its equivalent representation in terms of sine and cosine. This substitution is a crucial step because it brings all the trigonometric functions in the expression to a common denominator, which is essential for further simplification. It’s like translating everything into the same language so you can understand it better.

So, let's make the substitution. Our original expression is:

1 - sin(x) * cos(x) * tan(x)

Replacing tan(x) with sin(x) / cos(x), we get:

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

Notice how we've now expressed everything in terms of sine and cosine. This is a common strategy in trigonometric simplification because sine and cosine are fundamental functions, and we have powerful identities that relate them. The next step involves simplifying this new expression, which will bring us closer to our final answer. This step effectively sets the stage for the simplification process by converting the expression into a more manageable form. Keep an eye out for cancellations and common factors in the next step!

Step 2: Simplifying the Expression

Now that we've rewritten the expression as 1 - sin(x) * cos(x) * (sin(x) / cos(x)), the next step is to simplify it. This involves looking for opportunities to cancel out common factors and combine terms. In this case, we can see that cos(x) appears both in the numerator and the denominator of the second term. This is a classic opportunity for cancellation, and it's what makes this step so satisfying! It’s like watching the pieces of a puzzle fall into place.

Let's perform the cancellation. We have cos(x) in the numerator and cos(x) in the denominator, so they cancel each other out:

1 - sin(x) * ~~cos(x)~~ * (sin(x) / ~~cos(x)~~)

This leaves us with:

1 - sin(x) * sin(x)

Which can be written more concisely as:

1 - sin²(x)

We've successfully simplified the expression by canceling out the common factor. Notice how much cleaner the expression looks now! But we're not quite done yet. The simplified expression, 1 - sin²(x), looks very familiar, and that's because it's directly related to one of our fundamental trigonometric identities. In the next step, we'll use this identity to further simplify the expression and arrive at our final answer. Get ready to use your trigonometric toolkit one more time!

Step 3: Applying the Pythagorean Identity

We've arrived at the expression 1 - sin²(x), and it's time to use our knowledge of trigonometric identities to simplify it further. This is where the Pythagorean identity comes into play. Remember the identity: sin²(x) + cos²(x) = 1. This powerful identity can be rearranged to express cos²(x) in terms of sin²(x), and that's exactly what we're going to do here. It’s like using a secret code to unlock the final answer.

If we subtract sin²(x) from both sides of the Pythagorean identity, we get:

cos²(x) = 1 - sin²(x)

Now, look at our simplified expression: 1 - sin²(x). It's exactly the same as cos²(x)! This means we can directly substitute cos²(x) for 1 - sin²(x).

So, the final simplified form of the expression is:

cos²(x)

We've successfully used the Pythagorean identity to transform the expression into its simplest form. This step demonstrates the power of trigonometric identities in simplifying complex expressions. By recognizing the relationship between 1 - sin²(x) and cos²(x), we were able to reduce the expression to a single term. This is a testament to the elegance and interconnectedness of trigonometry. Now, let's take a look at the answer choices and see which one matches our simplified result!

Choosing the Correct Answer

Now that we've simplified the expression 1 - sin(x) * cos(x) * tan(x) to cos²(x), it's time to compare our result with the given answer choices. This step is crucial to ensure we've arrived at the correct solution. It’s like checking your work to make sure everything adds up.

Let's revisit the answer choices:

A) 1 - sin(x)

B) cos(x)

C) 1 - sin²(x)

D) 1 - sin(x) * sin(x) / cos(x)

We can see that answer choice C, 1 - sin²(x), is an intermediate step in our simplification process. While it's not incorrect, we simplified it further using the Pythagorean identity. However, the most simplified form we arrived at was cos²(x). So, none of the provided answer choices directly match our final simplified expression, cos²(x). But, we know that 1 - sin²(x) is equivalent to cos²(x).

Therefore, the correct answer is implicitly represented by C) 1 - sin²(x), as it is equivalent to our simplified result of cos²(x).

It's important to be aware that sometimes answer choices might not be in their most simplified form, and you need to recognize the equivalence between different expressions. This question highlights the importance of not only simplifying the expression but also understanding the relationships between different trigonometric forms. You nailed it!

Justification and Explanation of the Simplification Process

Let's recap the entire simplification process to solidify our understanding. This justification is crucial because it not only confirms our answer but also reinforces the underlying concepts and techniques. Think of it as summarizing the key takeaways from our journey.

Original Expression: We started with the expression 1 - sin(x) * cos(x) * tan(x).
Tangent Identity: We used the identity tan(x) = sin(x) / cos(x) to rewrite the expression as 1 - sin(x) * cos(x) * (sin(x) / cos(x)). This substitution was key to bringing all the trigonometric functions into terms of sine and cosine.
Simplification by Cancellation: We canceled out the common factor of cos(x) in the numerator and denominator, which simplified the expression to 1 - sin(x) * sin(x), or 1 - sin²(x).
Pythagorean Identity: We recognized that 1 - sin²(x) is equivalent to cos²(x) based on the Pythagorean identity sin²(x) + cos²(x) = 1. This allowed us to further simplify the expression to cos²(x).
Answer Choice: We compared our simplified result, cos²(x), with the given answer choices and determined that C) 1 - sin²(x) is the correct answer because it is equivalent to cos²(x).

Throughout this process, we used fundamental trigonometric identities and algebraic manipulation to simplify the expression. The key to success in trigonometric simplification is recognizing these identities and applying them strategically. By breaking down the problem into smaller steps and understanding the reasoning behind each step, we were able to confidently arrive at the correct answer. You're a trigonometry whiz!

Conclusion

Simplifying trigonometric expressions might seem challenging at first, but by understanding the fundamental identities and following a step-by-step approach, you can master the art of simplification. In this article, we tackled the expression 1 - sin(x) * cos(x) * tan(x) and successfully simplified it to cos²(x), which is equivalent to the answer choice C) 1 - sin²(x). Remember, the key is to break down the problem, use the appropriate identities, and simplify step by step. With practice and a solid understanding of trigonometric principles, you'll be simplifying expressions like a pro in no time! Keep up the great work, and don't hesitate to tackle more trigonometric challenges. You've got this! Now go out there and conquer those trigonometric expressions!