Hey guys! Today, we're diving into the fascinating world of trigonometry to tackle a classic problem: proving a trigonometric identity. Specifically, we'll be working on demonstrating that 1 + sin(a) / (1 + sin(a)) = sec(a)tan(a). This might look a bit intimidating at first glance, but don't worry! We'll break it down step by step, making sure everyone understands the logic behind each move. So, grab your calculators (though we won't really need them for this!), and let's get started on this trigonometric adventure!

Understanding the Basics

Before we jump into the proof itself, let's quickly refresh our understanding of some fundamental trigonometric concepts. This will ensure we're all on the same page and have the necessary tools to tackle the problem effectively. Remember, trigonometry is all about the relationships between angles and sides in triangles, and these relationships are expressed through various trigonometric functions.

Trigonometric Functions: A Quick Recap

The core trigonometric functions we'll be dealing with are sine (sin), cosine (cos), tangent (tan), secant (sec), and cosecant (csc). These functions are defined based on the ratios of sides in a right-angled triangle. Let's briefly define them:

• Sine (sin): The ratio of the opposite side to the hypotenuse.
• Cosine (cos): The ratio of the adjacent side to the hypotenuse.
• Tangent (tan): The ratio of the opposite side to the adjacent side. It can also be expressed as sin(a) / cos(a).
• Secant (sec): The reciprocal of cosine, meaning sec(a) = 1 / cos(a).
• Cosecant (csc): The reciprocal of sine, meaning csc(a) = 1 / sin(a).
• Cotangent (cot): The reciprocal of tangent, meaning cot(a) = 1 / tan(a) = cos(a) / sin(a).

Understanding these definitions is crucial because they form the building blocks of all trigonometric identities and proofs. If you're a little rusty on these, it might be a good idea to review them briefly before moving forward.

Key Trigonometric Identities

Besides the basic definitions, we also need to be familiar with some fundamental trigonometric identities. These identities are equations that are always true for any value of the angle (with some exceptions for values where the functions are undefined). The most important identity for this proof, and for trigonometry in general, is the Pythagorean identity:

• Pythagorean Identity: sin²(a) + cos²(a) = 1

This identity is derived from the Pythagorean theorem and is the cornerstone of many trigonometric manipulations. We'll also be using the definitions of secant and tangent, which we already covered. Knowing these identities and definitions is like having the right tools in your toolbox – they allow you to manipulate expressions and transform them into different forms.

Setting Up the Proof

Now that we've got our trigonometric toolkit ready, let's set the stage for proving the identity. Remember, our goal is to show that 1 + sin(a) / (1 + sin(a)) = sec(a)tan(a). The standard approach for proving identities is to start with one side of the equation (usually the more complex side) and manipulate it using known identities and algebraic techniques until it matches the other side.

Choosing the Starting Side

In this case, the right-hand side, sec(a)tan(a), looks like a good place to begin. Why? Because it involves secant and tangent, which can be expressed in terms of sine and cosine. This gives us a pathway to potentially simplify the expression and connect it to the left-hand side. The left-hand side, while seemingly simple, doesn't offer as many immediate avenues for manipulation.

The Initial Strategy

Our initial strategy will be to express sec(a) and tan(a) in terms of sine and cosine. This will allow us to work with a common set of trigonometric functions and hopefully reveal some opportunities for simplification. Remember, sec(a) = 1 / cos(a) and tan(a) = sin(a) / cos(a). By substituting these definitions into the right-hand side, we can rewrite the expression and start the process of transforming it.

Step-by-Step Proof

Alright, let's get down to the nitty-gritty and walk through the proof step by step. Remember, the key is to be methodical and show each step clearly. This not only helps you keep track of your progress but also makes it easier for others to follow your reasoning. We'll start with the right-hand side of the equation and work our way towards the left-hand side.

Step 1: Express sec(a) and tan(a) in terms of sin(a) and cos(a)

As we discussed earlier, the first step is to substitute the definitions of secant and tangent in terms of sine and cosine. This gives us:

sec(a)tan(a) = (1 / cos(a)) * (sin(a) / cos(a))

This substitution is crucial because it allows us to work with a common set of trigonometric functions, sine and cosine. Now, we have an expression that involves only sin(a) and cos(a), which opens up possibilities for further simplification.

Step 2: Simplify the expression

Now that we have everything in terms of sine and cosine, we can simplify the expression by multiplying the fractions:

(1 / cos(a)) * (sin(a) / cos(a)) = sin(a) / cos²(a)

This step is a straightforward application of fraction multiplication. We simply multiply the numerators and the denominators. The result is a single fraction with sin(a) in the numerator and cos²(a) in the denominator. We're making progress towards our goal!

Step 3: Use the Pythagorean Identity

This is where the Pythagorean identity comes into play. We know that sin²(a) + cos²(a) = 1. We can rearrange this identity to express cos²(a) in terms of sin²(a):

cos²(a) = 1 - sin²(a)

Now, we can substitute this expression for cos²(a) in our equation:

sin(a) / cos²(a) = sin(a) / (1 - sin²(a))

This substitution is a clever move because it allows us to introduce a term that resembles the left-hand side of our original equation. We're getting closer!

Step 4: Factor the denominator

The denominator, 1 - sin²(a), is a difference of squares, which can be factored as follows:

1 - sin²(a) = (1 - sin(a))(1 + sin(a))

This factorization is a key step in simplifying the expression further. By factoring the denominator, we create an opportunity to potentially cancel out a term in the numerator.

Step 5: Rewrite the expression with the factored denominator

Now, let's substitute the factored form of the denominator back into our equation:

sin(a) / (1 - sin²(a)) = sin(a) / ((1 - sin(a))(1 + sin(a)))

Step 6: Multiply the numerator and denominator by (1 - sin(a))

To manipulate the expression further and get it closer to our desired form, we can multiply both the numerator and the denominator by (1 - sin(a)):

[sin(a) / ((1 - sin(a))(1 + sin(a)))] * [(1 - sin(a)) / (1 - sin(a))] = sin(a) (1 - sin(a)) / ((1 - sin(a))(1 + sin(a))(1 - sin(a)))

Step 7: Expand the numerator

Expanding the numerator, we get:

sin(a) - sin²(a) / ((1 - sin(a))(1 + sin(a))(1 - sin(a)))

Step 8: Recognize that we made an error in our strategic approach

Okay, guys, it looks like we hit a bit of a snag here. If we continue down this road, we're not going to reach the desired result of (1 + sin(a)) / (1 + sin(a)). That's totally okay! In math, sometimes the path you initially choose doesn't lead to the destination, and you need to re-evaluate your strategy. It's part of the problem-solving process. Let's take a step back and analyze where we went wrong and how we can adjust our approach. This is a crucial skill in mathematics – being able to recognize when a strategy isn't working and being flexible enough to try something new.

Looking back, the step where we multiplied the numerator and denominator by (1 - sin(a)) led us down a more complicated path. While mathematically valid, it didn't simplify the expression in a way that brought us closer to our goal. Instead of trying to force a particular form, let's go back to the basics and think about what we're trying to achieve. We want to show that our expression is equivalent to (1 + sin(a)) / (1 + sin(a)), which is simply equal to 1. Maybe there's a more direct way to demonstrate this.

So, let's rewind to Step 5:

sin(a) / (1 - sin²(a)) = sin(a) / ((1 - sin(a))(1 + sin(a)))

And try a different approach from here.

Step 6 (Revised): Multiply the numerator and denominator by cos(a)

Instead of multiplying by (1-sin(a)), let's try multiplying both the numerator and the denominator by cos(a). This might seem a bit out of the blue, but we're looking for a way to introduce cosine terms in the numerator, which could potentially help us link back to the definitions of secant and tangent.

[sin(a) / ((1 - sin(a))(1 + sin(a)))] * [cos(a) / cos(a)] = sin(a)cos(a) / (((1 - sin(a))(1 + sin(a)))cos(a))

Step 7 (Revised): Substitute back cos²(a) for (1 - sin²(a))

Remember that (1 - sin²(a)) is the same as cos²(a). Let's make that substitution:

sin(a)cos(a) / (cos²(a) * cos(a)) = sin(a)cos(a) / cos³(a)

Step 8 (Revised): Simplify by canceling a common factor

We can cancel out a factor of cos(a) from the numerator and the denominator:

sin(a)cos(a) / cos³(a) = sin(a) / cos²(a)

Wait a minute... This looks familiar! We actually arrived back at an expression we had in our initial attempt (Step 2). This is a good reminder that sometimes going in a circle is part of the process. Even though this path didn't directly lead us to the answer, it does reinforce our earlier steps and perhaps gives us a new perspective.

Rethinking the Goal

Let's take another breather and really think about what we're trying to prove. We want to show that sec(a)tan(a) is equal to (1 + sin(a)) / (1 + sin(a)), which simplifies to 1. We've been manipulating the right-hand side, but maybe we need to consider if there's something we're missing about the left-hand side. Is there a way to manipulate the expression (1 + sin(a)) / (1 + sin(a)) to reveal its connection to secant and tangent?

The Revelation!

Okay, guys, I think I see the light! We've been so focused on manipulating the right-hand side that we've overlooked a crucial detail. The left-hand side, (1 + sin(a)) / (1 + sin(a)), is actually a bit of a trick! Any number (except 0) divided by itself is equal to 1. So, the left-hand side simplifies to 1.

This means our goal is now to show that sec(a)tan(a) = 1.

Why didn't we see this earlier? It's a classic case of getting caught up in the complexity of the trigonometric functions and missing the simple algebraic truth. This is a great lesson in problem-solving: always look for the simplest solution first!

Back to the Right Track

Now that we know our target is 1, let's revisit our previous work and see if we can get there. We left off at:

sin(a) / cos²(a)

We need to somehow manipulate this to equal 1. This is where another fundamental trigonometric identity comes into play.

Step 9 (Revised): Using the reciprocal identity

Let's rewrite the expression by separating out 1/cos(a):

sin(a) / cos²(a) = [sin(a) / cos(a)] * [1 / cos(a)]

Now, we can recognize the terms we've created:

• sin(a) / cos(a) = tan(a)
• 1 / cos(a) = sec(a)

Substituting these back into the equation, we get:

[sin(a) / cos(a)] * [1 / cos(a)] = tan(a)sec(a)

Step 10 (Revised): Final realization

Aha! We've gone full circle! We started with sec(a)tan(a) and, after some twists and turns, we've arrived back at sec(a)tan(a). But remember, our goal is to prove that this is equal to 1, which is what the left side simplifies to. We have made an error in the original problem statement, guys.

Conclusion

Woof! What a journey! We've been through a rollercoaster of trigonometric manipulations, faced some dead ends, and learned some valuable lessons about problem-solving along the way. While we didn't manage to prove that sec(a)tan(a) = (1 + sin(a)) / (1 + sin(a)), which simplifies to 1. Instead, we have understood why this is not an identity.

This exercise highlights the importance of:

• Understanding the fundamentals: Knowing the definitions of trigonometric functions and key identities is crucial.
• Strategic thinking: Choosing the right approach and adapting when necessary is key to success.
• Persistence: Don't give up when you hit a roadblock. Take a step back, re-evaluate, and try again.
• Attention to detail: Sometimes the simplest solution is the easiest to overlook.

Trigonometry can be challenging, but it's also incredibly rewarding. Keep practicing, keep exploring, and don't be afraid to make mistakes – they're just opportunities to learn! And hey, if you ever get stuck on a problem, remember this journey and the importance of perseverance. You got this!