Verify Trig Identity: (csc(x) + Cot(x)) / (sin(x) + Tan(x))

by Omar Yusuf 60 views

Hey guys! Let's dive into this trigonometric identity and see if it holds up. We've got:

$ \frac{ \csc(x) + \cot(x) }{ \sin(x) + \tan(x) } = \csc(x) \times \cot(x) $

It looks like a fun puzzle, so let’s break it down step by step. We'll start by rewriting everything in terms of sine and cosine, which will make things a lot clearer and easier to manipulate. Trust me, it’s like translating a foreign language – once you get the basic terms down, the rest starts to make sense.

Step 1: Rewrite in Terms of Sine and Cosine

Okay, so the first thing we need to do is express csc(x), cot(x), sin(x), and tan(x) using sin(x) and cos(x). Remember these fundamental identities? They're like the ABCs of trigonometry:

  • $ \csc(x) = \frac{1}{\sin(x)} $
  • $ \cot(x) = \frac{\cos(x)}{\sin(x)} $
  • $ \sin(x) = \sin(x) $ (This one's already in the form we want!)
  • $ \tan(x) = \frac{\sin(x)}{\cos(x)} $

Now, let's plug these into our original equation. It’s like swapping out ingredients in a recipe to see if we can make the same dish:

$ \frac{ \frac{1}{\sin(x)} + \frac{\cos(x)}{\sin(x)} }{ \sin(x) + \frac{\sin(x)}{\cos(x)} } = \frac{1}{\sin(x)} \times \frac{\cos(x)}{\sin(x)} $

This looks a bit messy, but don’t worry! We’re going to clean it up in the next step. The key here is to take it one piece at a time. Think of it like untangling a knot – patience is your best friend.

Step 2: Simplify the Numerator and Denominator

Alright, let's tackle the numerator first. We've got $ \frac{1}{\sin(x)} + \frac{\cos(x)}{\sin(x)} $. Since they both have the same denominator, we can combine them easily. It's like adding fractions you learned way back in grade school:

$ \frac{1 + \cos(x)}{\sin(x)} $

Great! Now, let's move on to the denominator, which is $ \sin(x) + \frac{\sin(x)}{\cos(x)} $. To combine these, we need a common denominator, which will be cos(x). So, we rewrite sin(x) as $ \frac{\sin(x)\cos(x)}{\cos(x)} $:

$ \frac{\sin(x)\cos(x)}{\cos(x)} + \frac{\sin(x)}{\cos(x)} $

Now we can add them:

$ \frac{\sin(x)\cos(x) + \sin(x)}{\cos(x)} $

We can even factor out a sin(x) from the numerator, which will help simplify things further. Factoring is like finding the common thread in a tangled mess of yarn:

$ \frac{\sin(x)(\cos(x) + 1)}{\cos(x)} $

So, our equation now looks like this:

$ \frac{ \frac{1 + \cos(x)}{\sin(x)} }{ \frac{\sin(x)(\cos(x) + 1)}{\cos(x)} } = \frac{1}{\sin(x)} \times \frac{\cos(x)}{\sin(x)} $

See? We're making progress! It might look complicated, but we’re just breaking it down into smaller, manageable pieces.

Step 3: Divide the Fractions

Okay, we've got a fraction divided by another fraction. The golden rule here is: dividing by a fraction is the same as multiplying by its reciprocal. Remember that? It's like flipping a pancake – you just turn it over:

So, we take the denominator $ \frac{\sin(x)(\cos(x) + 1)}{\cos(x)} $ and flip it to get $ \frac{\cos(x)}{\sin(x)(\cos(x) + 1)} $.

Now, we multiply the numerator by this reciprocal:

$ \frac{1 + \cos(x)}{\sin(x)} \times \frac{\cos(x)}{\sin(x)(\cos(x) + 1)} $

This is starting to look promising! We've got some terms that are just begging to be canceled out.

Step 4: Simplify by Canceling Terms

Here comes the fun part – canceling out common terms! We see that we have (1 + cos(x)) in both the numerator and the denominator, so we can cancel them out. It's like when you find matching socks in the laundry – so satisfying!

$ \frac{\cancel{(1 + \cos(x))}}{\sin(x)} \times \frac{\cos(x)}{\sin(x)\cancel{(\cos(x) + 1)}} $

This leaves us with:

$ \frac{1}{\sin(x)} \times \frac{\cos(x)}{\sin(x)} $

Now, let's look at the right side of our original equation:

$ \csc(x) \times \cot(x) $

We already know that $ \csc(x) = \frac{1}{\sin(x)} $ and $ \cot(x) = \frac{\cos(x)}{\sin(x)} $, so we can rewrite the right side as:

$ \frac{1}{\sin(x)} \times \frac{\cos(x)}{\sin(x)} $

Step 5: Compare Both Sides

Drumroll, please! Let's see what we've got. On the left side, we simplified our expression to:

$ \frac{1}{\sin(x)} \times \frac{\cos(x)}{\sin(x)} $

And on the right side, we have:

$ \frac{1}{\sin(x)} \times \frac{\cos(x)}{\sin(x)} $

They're the same! It’s like finding the last piece of a jigsaw puzzle – everything fits perfectly.

Conclusion: The Identity Holds True!

So, guys, after all that simplification and manipulation, we've shown that:

$ \frac{ \csc(x) + \cot(x) }{ \sin(x) + \tan(x) } = \csc(x) \times \cot(x) $

is indeed a valid trigonometric identity. We did it! This just goes to show that even complicated-looking equations can be solved if you break them down into smaller, more manageable steps. Keep practicing, and you'll be a trig whiz in no time!

Remember, the key is to:

  1. Rewrite everything in terms of sine and cosine.
  2. Simplify the numerator and denominator.
  3. Divide the fractions.
  4. Cancel out common terms.
  5. Compare both sides.

And most importantly, don't be afraid to get your hands dirty and play around with the equations. That's how you really learn and understand these things. Happy trig-ing!

Additional Insights and Tips for Trigonometric Identities

Let's delve a bit deeper into the fascinating world of trigonometric identities, guys! Now that we've successfully verified the given identity, it's a great time to explore some additional insights and tips that can help you tackle similar problems with confidence. Think of this as leveling up your trig skills!

Understanding the Foundation: Fundamental Identities

Before you even think about tackling complex identities, it’s crucial to have a rock-solid understanding of the fundamental trigonometric identities. These are your basic tools, like a hammer and screwdriver in a toolbox. Without them, you're going to have a tough time building anything. We touched on a few earlier, but let’s recap and expand a bit:

  • Reciprocal Identities: These define the reciprocals of the basic trig functions:
    • $ \csc(x) = \frac{1}{\sin(x)} $
    • $ \sec(x) = \frac{1}{\cos(x)} $
    • $ \cot(x) = \frac{1}{\tan(x)} $
  • Quotient Identities: These express tangent and cotangent in terms of sine and cosine:
    • $ \tan(x) = \frac{\sin(x)}{\cos(x)} $
    • $ \cot(x) = \frac{\cos(x)}{\sin(x)} $
  • Pythagorean Identities: These are derived from the Pythagorean theorem and are super useful:
    • $ \sin^2(x) + \cos^2(x) = 1 $
    • $ 1 + \tan^2(x) = \sec^2(x) $
    • $ 1 + \cot^2(x) = \csc^2(x) $

Mastering these identities is like learning the scales on a musical instrument. You need them ingrained in your memory so you can use them fluently.

Strategic Approaches to Verifying Identities

When you're faced with an identity to verify, there's no one-size-fits-all approach, but here are some strategies that can guide you:

  1. Start with the More Complicated Side: Usually, it’s easier to simplify a complex expression than to make a simple one more complex. Think of it like sculpting – you start with a block of clay and chip away at it to reveal the shape inside.
  2. Rewrite in Terms of Sine and Cosine: As we saw in our example, expressing everything in terms of sin(x) and cos(x) often reveals hidden cancellations and simplifications. It’s like speaking a common language – once everything is in the same terms, you can understand the relationships more clearly.
  3. Look for Pythagorean Substitutions: The Pythagorean identities are your best friends when it comes to simplifying squared trig functions. Keep an eye out for opportunities to use them. It’s like having a secret weapon in your arsenal!
  4. Combine Fractions: If you have fractions, try to combine them using a common denominator. This often leads to further simplification. It’s like merging two streams into a river – sometimes, the combined flow is more powerful.
  5. Factor Expressions: Factoring can reveal common factors that can be canceled out. It’s like finding the common thread in a tangled mess – once you pull it, the whole thing can unravel.
  6. Multiply by a Clever Form of 1: Sometimes, multiplying by a fraction like $ \frac{1 + \cos(x)}{1 + \cos(x)} $ can help you create Pythagorean identities or other useful forms. This might seem like a magic trick, but it’s a valid algebraic manipulation because you’re essentially multiplying by 1.

Common Mistakes to Avoid

Trigonometric identities can be tricky, and it’s easy to make mistakes if you’re not careful. Here are a few common pitfalls to watch out for:

  • Don't Treat It Like an Equation to Solve: You're not solving for x here. You're trying to show that two expressions are equivalent. This means you should work on each side separately until they match.
  • Avoid Illegal Operations: Be careful not to perform operations that change the value of the expression, like adding something to only one side or taking the square root of only one side.
  • Double-Check Your Algebra: A lot of mistakes in trig identities come down to simple algebraic errors. Take your time and double-check your work.

Practice, Practice, Practice!

The best way to get good at verifying trigonometric identities is to practice. The more problems you solve, the more comfortable you'll become with the techniques and strategies. It’s like learning any new skill – the more you do it, the better you get.

  • Start with Simpler Identities: Build your confidence by tackling easier problems first.
  • Work Through Examples: Study worked examples carefully to see how others approach the problems.
  • Don't Give Up: Some identities can be challenging, but don't get discouraged. Keep trying, and you'll eventually get it.

Wrapping Up: Embracing the Beauty of Trigonometry

Trigonometric identities might seem like abstract mathematical puzzles, but they're incredibly powerful tools that have applications in many areas of science and engineering. Understanding them deeply is not just about getting good grades; it's about developing a powerful problem-solving mindset.

So, keep exploring, keep practicing, and most importantly, keep having fun with trigonometry, guys! You've got this!