Prove: (1+sin(x))/(1-sin(x)) - (1-sin(x))/(1+sin(x)) = 4tan(x)sec(x)

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Prove: (1+sin(x))/(1-sin(x)) - (1-sin(x))/(1+sin(x)) = 4tan(x)sec(x)

Hey guys! Let's dive into proving this cool trigonometric identity. It looks a bit intimidating at first, but we'll break it down step-by-step. Our mission is to show that the expression on the left side of the equation, which is (1+sin(x))/(1-sin(x)) - (1-sin(x))/(1+sin(x)), is actually the same as the expression on the right side, which is 4tan(x)sec(x). To do this, we will start with the left side and use algebraic manipulations and trigonometric identities to transform it until it looks exactly like the right side. It's like a mathematical puzzle, and we're about to solve it!

Starting with the Left Side: Fraction Subtraction

Okay, so the first thing we're going to tackle is the left side of the equation. We've got two fractions being subtracted, and the key here is to combine them into a single fraction. Remember how we do that? We need a common denominator! In this case, our denominators are (1 - sin(x)) and (1 + sin(x)). So, the least common denominator is simply their product, which is (1 - sin(x))(1 + sin(x)). Now, let's get those fractions singing the same tune.

To get the common denominator, we'll multiply the first fraction, (1+sin(x))/(1-sin(x)), by (1+sin(x))/(1+sin(x)). This doesn't change the value of the fraction because we're essentially multiplying by 1. Similarly, we'll multiply the second fraction, (1-sin(x))/(1+sin(x)), by (1-sin(x))/(1-sin(x)). This gives us:

[(1 + sin(x))(1 + sin(x))] / [(1 - sin(x))(1 + sin(x))]  -  [(1 - sin(x))(1 - sin(x))] / [(1 + sin(x))(1 - sin(x))]

Now we have a common denominator, which is awesome! Let's simplify those numerators by multiplying out the binomials.

Expanding the Numerators

Let's take a closer look at those numerators. We've got (1 + sin(x))(1 + sin(x)) and (1 - sin(x))(1 - sin(x)). These are just binomials being multiplied by themselves, so we can use the FOIL method (First, Outer, Inner, Last) or the perfect square trinomial pattern to expand them.

For (1 + sin(x))(1 + sin(x)), we get:

1 + sin(x) + sin(x) + sinĀ²(x) = 1 + 2sin(x) + sinĀ²(x)

And for (1 - sin(x))(1 - sin(x)), we get:

1 - sin(x) - sin(x) + sinĀ²(x) = 1 - 2sin(x) + sinĀ²(x)

Now we can substitute these expanded forms back into our main expression:

[1 + 2sin(x) + sinĀ²(x)] / [(1 - sin(x))(1 + sin(x))]  -  [1 - 2sin(x) + sinĀ²(x)] / [(1 + sin(x))(1 - sin(x))]

We're making progress, guys! Next, let's simplify the denominator and then combine the fractions.

Simplifying the Denominator and Combining Fractions

Alright, let's focus on the denominator. We've got (1 - sin(x))(1 + sin(x)). This looks like the difference of squares pattern, which is (a - b)(a + b) = aĀ² - bĀ². So, we can simplify this to 1Ā² - sinĀ²(x), which is just 1 - sinĀ²(x). Remember that Pythagorean identity, sinĀ²(x) + cosĀ²(x) = 1? We can rearrange it to get cosĀ²(x) = 1 - sinĀ²(x). How cool is that? Our denominator simplifies beautifully to cosĀ²(x)!

Now our expression looks like this:

[1 + 2sin(x) + sinĀ²(x)] / cosĀ²(x)  -  [1 - 2sin(x) + sinĀ²(x)] / cosĀ²(x)

Since we have a common denominator, we can finally combine the fractions. We subtract the numerators, being careful to distribute the negative sign:

[1 + 2sin(x) + sinĀ²(x) - (1 - 2sin(x) + sinĀ²(x))] / cosĀ²(x)

Distributing the negative sign, we get:

[1 + 2sin(x) + sinĀ²(x) - 1 + 2sin(x) - sinĀ²(x)] / cosĀ²(x)

Now let's simplify by combining like terms. The 1 and -1 cancel out, and the sinĀ²(x) and -sinĀ²(x) cancel out too. We're left with:

[4sin(x)] / cosĀ²(x)

We're getting closer to our goal! Now we need to massage this expression into the form 4tan(x)sec(x). Let's see how we can do that.

Transforming to the Right Side: Using Trigonometric Identities

Okay, we've simplified the left side to 4sin(x) / cosĀ²(x). Now the fun part: let's transform this into 4tan(x)sec(x). Remember our definitions for tangent and secant? tan(x) = sin(x) / cos(x) and sec(x) = 1 / cos(x). We need to somehow get these terms to appear in our expression.

Notice that we have cosĀ²(x) in the denominator. We can think of this as cos(x) * cos(x). Let's rewrite our expression like this:

4sin(x) / [cos(x) * cos(x)]

Now, let's separate out one of the cos(x) terms to form our tangent:

4 * [sin(x) / cos(x)] * [1 / cos(x)]

See what we did there? We've cleverly isolated sin(x) / cos(x), which we know is tan(x). And we also have 1 / cos(x), which is sec(x). Substituting these in, we get:

4 * tan(x) * sec(x)

And that's exactly what we wanted to show! We've successfully transformed the left side of the equation into the right side.

Conclusion: We Did It!

Alright guys, we did it! We started with the left side of the identity, (1+sin(x))/(1-sin(x)) - (1-sin(x))/(1+sin(x)), and through a series of algebraic manipulations and trigonometric substitutions, we arrived at the right side, 4tan(x)sec(x). This proves that the identity is true.

To recap, we:

  1. Found a common denominator and combined the fractions.
  2. Expanded the numerators and simplified the denominator using the difference of squares pattern and the Pythagorean identity.
  3. Combined like terms and simplified the expression.
  4. Used the definitions of tangent and secant to rewrite the expression in the desired form.

This was a fantastic exercise in using our trigonometric toolboxes! Keep practicing these kinds of problems, and you'll become a trig identity master in no time. Great job, everyone! Remember, the key is to break down complex problems into smaller, manageable steps and to utilize the fundamental identities we've learned. Keep up the awesome work! This trigonometric identity proof showcases the power of algebraic manipulation combined with trigonometric identities. By systematically working through each step, we successfully transformed a complex expression into a simpler, equivalent form. The use of the common denominator, difference of squares, and the fundamental Pythagorean identity were crucial in this process. Furthermore, recognizing the definitions of tangent and secant allowed us to make the final connection and complete the proof. This exercise highlights the importance of having a strong foundation in both algebra and trigonometry to tackle these types of problems effectively. With consistent practice and a solid understanding of the core concepts, anyone can master the art of proving trigonometric identities. So, keep exploring, keep learning, and most importantly, keep having fun with math! Each time you solve a problem like this, you're not just getting the answer; you're building a deeper understanding of the mathematical relationships and sharpening your problem-solving skills. And that's what it's all about! Remember to always check your work and ensure each step is logically sound. This helps prevent errors and reinforces the understanding of the concepts. If you ever get stuck, don't hesitate to review the fundamental identities and algebraic techniques. There are also plenty of resources available online and in textbooks that can provide additional guidance and examples. The journey of learning mathematics is a continuous process, and each challenge is an opportunity to grow and improve. So, embrace the complexity, celebrate the breakthroughs, and never stop exploring the fascinating world of trigonometry and beyond! You've got this!