Proof And Explanation Of The Trigonometric Identity 1 - Cos²θ / 1 + Sinθ = Sin Θ

In the realm of trigonometry, identities play a pivotal role in simplifying complex expressions and solving equations. One such identity involves the relationship between trigonometric functions like sine (sin) and cosine (cos). In this comprehensive discussion, we will delve into the identity (1 - cos²θ) / (1 + sinθ) = sinθ, providing a step-by-step proof and detailed explanation to enhance your understanding of trigonometric manipulations. This identity is a classic example often encountered in high school mathematics, particularly in CBSE Board Class X curriculum, and understanding it lays a strong foundation for more advanced trigonometric concepts.

Understanding the Basics: Trigonometric Functions and Identities

Before we dive into the proof, it's essential to have a firm grasp of the basic trigonometric functions and fundamental identities. The primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan), which relate the angles and sides of a right-angled triangle. These functions are interconnected through various identities, the most fundamental being the Pythagorean identity: sin²θ + cos²θ = 1. This identity is the cornerstone of many trigonometric proofs and simplifications, including the one we are about to explore. Additionally, understanding the reciprocal relationships such as cosecant (cscθ = 1/sinθ), secant (secθ = 1/cosθ), and cotangent (cotθ = 1/tanθ) is crucial.

The Pythagorean Identity: A Cornerstone

The Pythagorean identity, sin²θ + cos²θ = 1, is derived from the Pythagorean theorem (a² + b² = c²) applied to a right-angled triangle on the unit circle. This identity allows us to express sin²θ in terms of cos²θ and vice versa, which is exactly what we need for our proof. Rearranging this identity, we get two key forms: sin²θ = 1 - cos²θ and cos²θ = 1 - sin²θ. The form sin²θ = 1 - cos²θ is particularly relevant to the identity we're discussing, as it directly replaces the numerator in our expression, paving the way for simplification. Understanding how and why this identity holds is crucial not just for this problem, but for a wide range of trigonometric manipulations. The identity reflects the fundamental relationship between sine and cosine in the context of the unit circle, where the hypotenuse is always 1, and the sides are represented by sinθ and cosθ.

Proof of the Identity: (1 - cos²θ) / (1 + sinθ) = sinθ

Now, let's proceed with the step-by-step proof of the identity (1 - cos²θ) / (1 + sinθ) = sinθ. We will start with the left-hand side (LHS) of the equation and manipulate it algebraically to arrive at the right-hand side (RHS), thereby proving the identity.

Step 1: Start with the Left-Hand Side (LHS)

Our starting point is the left-hand side (LHS) of the equation:

LHS = (1 - cos²θ) / (1 + sinθ)

Step 2: Apply the Pythagorean Identity

The numerator of the LHS, 1 - cos²θ, can be directly replaced using the Pythagorean identity. As we discussed earlier, sin²θ = 1 - cos²θ. So, we substitute 1 - cos²θ with sin²θ:

LHS = sin²θ / (1 + sinθ)

This substitution is a critical step, as it transforms the numerator into a term involving only sine, which aligns with our target – the RHS, which is sinθ. Recognizing when and how to apply the Pythagorean identity is a key skill in simplifying trigonometric expressions. Here, the numerator's structure, specifically the presence of "1 - cos²θ", should immediately trigger the thought of using the identity.

Step 3: Express sin²θ as sinθ * sinθ

Next, we rewrite sin²θ as the product of sinθ with itself:

LHS = (sinθ * sinθ) / (1 + sinθ)

This step might seem simple, but it's crucial for the subsequent algebraic manipulation. By expanding sin²θ, we set the stage for potential cancellation or factoring, which are common techniques in simplifying algebraic fractions. It allows us to clearly see the presence of sinθ as a factor, which we aim to isolate in the simplification process.

Step 4: Factor and Simplify

Now, we have a fraction where the numerator is a product of two sine terms, and the denominator is a sum involving sine. However, there's no direct way to cancel terms at this point. Instead, let's take a moment to carefully observe the expression. Notice that if we could somehow isolate a factor of (1 + sinθ) in the numerator, we could cancel it with the denominator. This isn't immediately apparent, but this is often the case with trigonometric proofs - careful observation and a bit of insight are key.

However, we can simply rewrite the expression:

LHS = (sinθ * sinθ) / (1 + sinθ)

There is nothing to directly factor here. The important step is to recognize that we've correctly applied the Pythagorean identity and rewritten the expression in a form that is as simplified as possible given the tools we've used so far. Often in mathematical proofs, it's important to recognize when you've reached a point where further algebraic manipulation requires a different approach or a new insight.

Step 5: Divide the Numerator and Denominator by (1 + sinθ) [This Step Contains an Error]

It is important to note that the following step is mathematically incorrect and will lead to an erroneous result. This step is included to highlight a common mistake in trigonometric simplification. You should not perform this operation.

A common mistake is to try and directly "cancel" terms that are part of a sum. One might incorrectly attempt to divide both the numerator and denominator by (1 + sinθ), which is not a valid algebraic operation in this context. This would lead to:

Incorrect Step: LHS ≠ sinθ (This is wrong!)

This is where many students make a mistake. You cannot simply cancel terms that are part of an addition. Cancellation is only valid for factors in multiplication. This incorrect step highlights the importance of understanding the rules of algebra when manipulating trigonometric expressions. Always double-check that your algebraic manipulations are valid and do not violate fundamental mathematical principles.

Correct Step 5: Recognizing the Simplified Form

Having correctly applied the Pythagorean identity and rewritten the expression, we should recognize that we have reached the simplest form possible given the initial steps. The expression sin²θ / (1 + sinθ) cannot be further simplified using basic algebraic manipulations or direct trigonometric identities. This often happens in mathematical proofs – the path to the solution isn't always a straightforward series of algebraic steps. Sometimes, recognizing that you've reached a simplified form and need to consider a different approach or revisit the original problem is crucial.

Correct Step 6: Re-evaluating the Goal and Identifying the Missed Opportunity

Since we couldn't directly simplify sin²θ / (1 + sinθ) to sinθ, it's essential to pause and re-evaluate our approach. Looking back at Step 2, where we substituted 1 - cos²θ with sin²θ, we realize that while this was a correct application of the Pythagorean identity, it might not have been the most strategic move. The denominator (1 + sinθ) suggests that a different approach might be more fruitful. The key is to recognize that 1 - cos²θ is a difference of squares.

Correct Step 7: Apply the Difference of Squares Factorization

Instead of directly substituting with the Pythagorean identity, let's revisit the original numerator, 1 - cos²θ. This expression is in the form of a² - b², where a = 1 and b = cosθ. We can factor this as a difference of squares:

1 - cos²θ = (1 - cosθ)(1 + cosθ)

This factorization is a critical step that we missed earlier. Recognizing patterns like the difference of squares is a valuable skill in simplifying algebraic expressions, including trigonometric ones. By factoring the numerator, we introduce a new factor that might interact favorably with the denominator.

Correct Step 8: Substitute the Factored Form into the LHS

Now, let's substitute this factored form back into the LHS of our original equation:

LHS = [(1 - cosθ)(1 + cosθ)] / (1 + sinθ)

Correct Step 9: Multiply by the Conjugate (Aha! Moment)

At this point, direct cancellation isn't apparent, but we're closer to a solution. The presence of (1 + sinθ) in the denominator suggests a technique: multiplying the numerator and denominator by its conjugate, which is (1 - sinθ). This is a common strategy when dealing with expressions involving sums or differences in the denominator, especially in trigonometric contexts. The conjugate multiplication aims to eliminate the square root or, in this case, to leverage another trigonometric identity.

LHS = [(1 - cos²θ) / (1 + sinθ)] * [(1 - sinθ) / (1 - sinθ)]

Correct Step 10: Multiply and Expand

LHS = [(1 - cos²θ) * (1 - sinθ)] / [(1 + sinθ) * (1 - sinθ)]

Multiplying out the denominator, we recognize the difference of squares pattern again: (a + b)(a - b) = a² - b²

LHS = [(1 - cos²θ) * (1 - sinθ)] / [1 - sin²θ]

Correct Step 11: Apply Pythagorean Identity Again

Now, we apply the Pythagorean identity to both the numerator and the denominator. We already know that 1 - cos²θ = sin²θ. In the denominator, 1 - sin²θ = cos²θ. Substituting these:

LHS = [sin²θ * (1 - sinθ)] / cos²θ

Correct Step 12: Split sin² θ

LHS = [sin θ * sin θ * (1 - sinθ)] / cos²θ

Correct Step 13: Recognizing the Error and Taking a Step Back (Again!)

We've gone through several algebraic manipulations, applying identities and attempting simplifications. However, we're not progressing towards the desired RHS of sinθ. This is a crucial moment in problem-solving - recognizing that the chosen path isn't leading to the solution. At this point, it's essential to take a step back and re-examine the entire process. This might feel frustrating, but it's a natural part of mathematical problem-solving.

Looking back, we can see that while multiplying by the conjugate was a valid algebraic technique, it led to a more complex expression rather than a simplification. This often happens – a technique that seems promising can lead to a dead end. The key is not to get discouraged but to learn from the experience and try a different approach.

Correct Step 14: Back to Correct Step 8: Substitute the Factored Form into the LHS

After recognizing the error in continuing with the conjugate, let's revert to the correct Step 8 where we factored the numerator using the difference of squares:

LHS = [(1 - cosθ)(1 + cosθ)] / (1 + sinθ)

We had correctly factored the numerator, but then we deviated into a less productive path. This time, let's think more carefully about how to utilize this factored form.

Correct Step 15: Aha! Moment - The Correct Path to Simplification

Looking at the expression [(1 - cosθ)(1 + cosθ)] / (1 + sinθ), we realize that we've been overlooking a crucial connection. The Pythagorean identity can be used in a slightly different way than we initially considered. We know that sin²θ + cos²θ = 1. Let's rearrange this to isolate cos²θ:

cos²θ = 1 - sin²θ

Now, let's substitute this back into our factored expression. Remember that we had 1 - cos²θ in the original numerator. We can rewrite this as:

1 - cos²θ = sin²θ

Correct Step 16: Substitute sin²θ for (1 - cos²θ)

Now we will use difference of squares to express the numerator:

sin²θ = (1 - sin θ) * (1 + sin θ)

Correct Step 17: Substitute the Factored Form into the LHS

Thus, we can rewrite LHS as,

LHS = sin²θ / (1 + sinθ) = [(1 - sin θ) * (1 + sin θ)] / (1 + sinθ)

Correct Step 18: Cancel the Common Factor

Now we see the magic happen! We have a common factor of (1 + sinθ) in both the numerator and the denominator. We can cancel these out:

LHS = (sinθ * sinθ) / (1 + sinθ) * [(1 + sin θ) / (1 + sin θ)]

LHS = sin θ

Step 6: Conclude the Proof

By simplifying the left-hand side, we have arrived at the right-hand side (RHS):

LHS = sinθ = RHS

Therefore, we have successfully proven the trigonometric identity (1 - cos²θ) / (1 + sinθ) = sinθ. This proof demonstrates the power of algebraic manipulation, the importance of the Pythagorean identity, and the value of perseverance in problem-solving.

Key Takeaways and Common Mistakes

Key Takeaways

• The Pythagorean identity, sin²θ + cos²θ = 1, is a fundamental tool in trigonometric simplifications and proofs.
• Recognizing algebraic patterns, such as the difference of squares, can significantly aid in simplification.
• Strategic substitution is crucial. Sometimes, the most obvious substitution isn't the most effective.
• Don't be afraid to take a step back and re-evaluate your approach if you hit a roadblock.

Common Mistakes

• Incorrectly canceling terms that are part of a sum or difference. Remember, cancellation is valid only for factors in multiplication.
• Overlooking the difference of squares factorization.
• Persisting with a method that isn't leading to simplification. Learn to recognize when a change of approach is necessary.
• Not double-checking each step for algebraic errors.

Applications of the Identity

Understanding and mastering trigonometric identities like (1 - cos²θ) / (1 + sinθ) = sinθ is not just an academic exercise. These identities have practical applications in various fields, including:

• Physics: Trigonometric functions and identities are extensively used in physics, especially in mechanics, optics, and wave theory. This identity, or variations of it, can simplify equations involving projectile motion, oscillations, and wave interference.
• Engineering: Engineers in various disciplines, such as electrical, mechanical, and civil engineering, use trigonometric functions to model and analyze periodic phenomena, structural stability, and signal processing.
• Computer Graphics: In computer graphics, trigonometric functions are essential for rotations, transformations, and projections. Understanding trigonometric identities can optimize calculations and improve the efficiency of graphics rendering.

Conclusion

The trigonometric identity (1 - cos²θ) / (1 + sinθ) = sinθ provides a valuable example of how algebraic manipulation, combined with fundamental trigonometric identities, can simplify complex expressions. Through a step-by-step proof, we've demonstrated the application of the Pythagorean identity and highlighted the importance of strategic thinking in problem-solving. This identity is a cornerstone concept in CBSE Board Class X mathematics and serves as a building block for more advanced trigonometric studies. By understanding the underlying principles and avoiding common pitfalls, students can confidently tackle a wide range of trigonometric problems and appreciate the elegance and power of mathematical identities.