Simplifying Trigonometric Expressions: A Step-by-Step Guide
Hey guys! Let's dive into the world of trigonometry and simplify a pretty cool expression. This is a common type of problem you might encounter in your math journey, so understanding how to tackle it is super valuable. We'll break down the expression step by step, making sure you grasp the concepts and techniques involved. Get ready to flex those math muscles! We're going to break down this expression: $ sinx + cosx \div { cos }^{2} x \div 1 + sinx - { sin}^{2} x \div 1 - cosx $. Our main goal is to find the most simplified form of this expression. Don't worry, it seems complicated at first glance, but with the right approach, it becomes manageable. Let's get started!
Understanding the Basics: Trigonometric Identities
Before we jump into the simplification, let's quickly recap some fundamental trigonometric identities. These identities are our best friends when it comes to simplifying trig expressions. They allow us to rewrite parts of the expression in different forms, making it easier to work with. Some of the key identities we'll use include the Pythagorean identity: $ { sin }^{2} x + { cos }^{2} x = 1 $. This is one of the most important ones! Also, remember that $ tan x = sin x / cos x $, $ cot x = cos x / sin x $, $ sec x = 1 / cos x $, and $ csc x = 1 / sin x $. Keep these in mind as we go through the simplification. They are super helpful. Learning these will give you a major advantage. These identities are the foundation of trigonometry, and mastering them will make your life a whole lot easier when dealing with complex expressions. Knowing them will help you transform and manipulate the expression in order to find a simplified form. Remember, the key is to look for opportunities to apply these identities. It is like having a secret weapon. The more you practice, the more familiar these identities become, and the easier it will be to spot where they can be used. It is like learning a new language. You start with the alphabet, then you learn to form words, and eventually, you can have full conversations. The same goes for these trigonometric identities; you begin with the basics and progressively build your understanding, leading you to master complex expressions. Now, let's put these identities into action.
Applying the Identities
Now, let's start simplifying the given expression step by step. Our goal is to manipulate the expression using trigonometric identities to arrive at a simpler form. Remember the original expression: $ sinx + cosx \div cos }^{2} x \div 1 + sinx - { sin}^{2} x \div 1 - cosx $. First, let's rewrite the division operations. Remember that dividing by 1 doesn't change anything, so we can rewrite the expression as $ sinx + cosx / { cos }^{2} x + sinx - { sin }^{2} x - cosx $. The most important thing here is to organize the different parts of the original expression. Now, we can simplify $ cosx / { cos }^{2} x $ to $ 1 / cos x $, provided that $ cos x $ is not equal to zero. This simplifies because one of the cosine terms in the denominator cancels out with the cosine term in the numerator. Also, we can rewrite $ 1 / cos x $ as $ sec x $. Thus, our expression now looks like this^{2} x - cosx $. See how the expression is already looking a bit cleaner? Great! We are making progress. Now is a great time to apply some of our identities. The key is to look for terms or groups of terms where identities can be applied. We can rearrange the terms. Group similar terms together to make it easier to see where we can apply identities or simplify further. For example, the expression can be rewritten as $ 2sinx - cosx - { sin }^{2} x + sec x $. At this point, it does not look like there are any obvious next steps, but with more practice, you'll start to recognize patterns. Patience is important! Now, we will consider another form to see if we can simplify the expression further.
Further Simplification and Final Answer
Okay, let's continue simplifying the expression. Remember, we have the expression $ 2sinx - cosx - sin }^{2} x + sec x $. There might not be a straightforward next step, but we can try to find common terms or factor expressions, but it does not appear that we can perform either. When simplifying trigonometric expressions, sometimes the simplest form isn't immediately obvious, and we have to try different approaches. We might rearrange the terms again to see if it helps us identify a way to simplify further. Let's try grouping the terms in a different way. Grouping the terms in different ways can reveal new possibilities for simplification. Let's try rearranging the expression as follows^{2} x) + (sinx - cosx + sec x) $. We are simply regrouping the terms here to see if we can identify any patterns or potential simplifications. Factoring out $ sin x $ from the first group gives us $ sinx(1 - sinx) + (sinx - cosx + secx) $. This doesn't seem to lead to any immediate simplification either, but sometimes, rearranging the terms can reveal a path forward that wasn't initially visible. The key is to be flexible and try different approaches until you find one that works. Trigonometric simplification often requires a bit of trial and error. You might try different identities or arrangements of the terms until you find a path to the simplest form. It is like solving a puzzle; you keep trying different pieces until they fit. Even if a path doesn't immediately lead to a solution, it provides valuable insights and helps you understand the expression better. With the steps we've taken, it appears that we have reached the most simplified form. We've used various identities, rearranged terms, and simplified the expression as much as possible. Therefore, the simplified form of $ sinx + cosx \div { cos }^{2} x \div 1 + sinx - { sin}^{2} x \div 1 - cosx $ is $ 2sinx - cosx - { sin }^{2} x + sec x $. This is a good example of how to tackle a complex expression in trigonometry.
The Final Result
So, after all the steps, the simplified form is: $ 2sinx - cosx - { sin }^{2} x + sec x $. This is the simplest form we could achieve by applying trigonometric identities and algebraic manipulations.
Remember, practice is key! The more you work with trigonometric expressions, the more comfortable and proficient you will become. Keep exploring, keep practicing, and you'll master these types of problems in no time. If you got this far, congrats! You've done a great job simplifying the expression. Always remember to double-check your work and to verify that your final answer is indeed the most simplified form. You can always plug in some values to see if your answer is consistent with the original expression. Now, go out there and keep simplifying! Keep practicing, and you'll get better with each problem you solve.