Prove Trigonometric Identity Cos^2(θ)/(1-tan(θ)) + Sin^3(θ)/(sin(θ)-cos(θ)) = 1 + Sin(θ)cos(θ)

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In this article, we will delve into the fascinating world of trigonometry and tackle the challenge of proving a specific trigonometric identity. Trigonometric identities are fundamental equations that hold true for all values of the variables involved. Mastering these identities is crucial for simplifying complex expressions, solving trigonometric equations, and gaining a deeper understanding of the relationships between trigonometric functions. We will focus on proving the following identity:

a) cos⁡2θ1−tan⁡θ+sin⁡3θsin⁡θ−cos⁡θ=1+sin⁡θcos⁡θ\frac{\cos ^2 \theta}{1-\tan \theta}+\frac{\sin ^3 \theta}{\sin \theta-\cos \theta}=1+\sin \theta \cos \theta

We will provide a comprehensive, step-by-step solution, breaking down each step and explaining the underlying trigonometric principles involved. This guide aims to make the process clear and accessible, even for those who are relatively new to trigonometry. Let's embark on this mathematical journey together!

Understanding Trigonometric Identities

Before we dive into the proof, let's take a moment to understand what trigonometric identities are and why they are so important. Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables for which the functions are defined. These identities are essential tools in trigonometry, calculus, and various branches of physics and engineering.

The basic trigonometric functions are sine (sin⁡θ\sin \theta), cosine (cos⁡θ\cos \theta), tangent (tan⁡θ\tan \theta), cosecant (csc⁡θ\csc \theta), secant (sec⁡θ\sec \theta), and cotangent (cot⁡θ\cot \theta). These functions relate the angles of a right triangle to the ratios of its sides. The fundamental trigonometric identities are derived from the definitions of these functions and the Pythagorean theorem.

The importance of trigonometric identities lies in their ability to simplify complex trigonometric expressions. By using identities, we can rewrite expressions in a more manageable form, making it easier to solve equations, evaluate integrals, and perform other mathematical operations. Moreover, these identities provide a deeper insight into the relationships between trigonometric functions, allowing us to understand their behavior and properties more effectively.

In the context of this article, we will utilize key identities such as the definition of tangent in terms of sine and cosine, and algebraic manipulations to demonstrate the given identity. Understanding these fundamental concepts is crucial for not only proving this specific identity but also for tackling a wide range of trigonometric problems.

Step-by-Step Proof of the Identity

Now, let's proceed with the proof of the identity. We will start by manipulating the left-hand side (LHS) of the equation and transforming it into the right-hand side (RHS). This approach is a common strategy in proving identities: we work with one side of the equation, applying known identities and algebraic manipulations, until it matches the other side.

Step 1: Rewrite the tangent function

Our first step involves rewriting the tangent function in terms of sine and cosine. Recall that tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}. Substituting this into the LHS, we get:

cos⁡2θ1−sin⁡θcos⁡θ+sin⁡3θsin⁡θ−cos⁡θ\frac{\cos ^2 \theta}{1-\frac{\sin \theta}{\cos \theta}}+\frac{\sin ^3 \theta}{\sin \theta-\cos \theta}

This substitution is a crucial first step as it allows us to express the entire equation in terms of sine and cosine, the two fundamental trigonometric functions. This simplifies the subsequent algebraic manipulations.

Step 2: Simplify the denominator of the first term

Next, we simplify the denominator of the first term. To do this, we find a common denominator for the expression 1−sin⁡θcos⁡θ1-\frac{\sin \theta}{\cos \theta}, which is cos⁡θ\cos \theta. This gives us:

1−sin⁡θcos⁡θ=cos⁡θ−sin⁡θcos⁡θ1-\frac{\sin \theta}{\cos \theta} = \frac{\cos \theta - \sin \theta}{\cos \theta}

Substituting this back into the LHS, we have:

cos⁡2θcos⁡θ−sin⁡θcos⁡θ+sin⁡3θsin⁡θ−cos⁡θ\frac{\cos ^2 \theta}{\frac{\cos \theta - \sin \theta}{\cos \theta}}+\frac{\sin ^3 \theta}{\sin \theta-\cos \theta}

This step is essential for combining the fractions and simplifying the overall expression. By finding a common denominator, we are able to rewrite the first term in a more manageable form.

Step 3: Divide by a fraction

Now, we simplify the first term by dividing by a fraction. Recall that dividing by a fraction is the same as multiplying by its reciprocal. Thus,

cos⁡2θcos⁡θ−sin⁡θcos⁡θ=cos⁡2θ⋅cos⁡θcos⁡θ−sin⁡θ=cos⁡3θcos⁡θ−sin⁡θ\frac{\cos ^2 \theta}{\frac{\cos \theta - \sin \theta}{\cos \theta}} = \cos ^2 \theta \cdot \frac{\cos \theta}{\cos \theta - \sin \theta} = \frac{\cos ^3 \theta}{\cos \theta - \sin \theta}

Substituting this back into the LHS, we get:

cos⁡3θcos⁡θ−sin⁡θ+sin⁡3θsin⁡θ−cos⁡θ\frac{\cos ^3 \theta}{\cos \theta - \sin \theta}+\frac{\sin ^3 \theta}{\sin \theta-\cos \theta}

This step is a key algebraic manipulation that simplifies the complex fraction and sets the stage for combining the two terms.

Step 4: Find a common denominator

Notice that the denominators of the two terms are almost the same, but with opposite signs. To obtain a common denominator, we multiply the numerator and denominator of the second term by -1:

sin⁡3θsin⁡θ−cos⁡θ=−sin⁡3θcos⁡θ−sin⁡θ\frac{\sin ^3 \theta}{\sin \theta-\cos \theta} = \frac{-\sin ^3 \theta}{\cos \theta - \sin \theta}

Now, the LHS becomes:

cos⁡3θcos⁡θ−sin⁡θ−sin⁡3θcos⁡θ−sin⁡θ\frac{\cos ^3 \theta}{\cos \theta - \sin \theta}-\frac{\sin ^3 \theta}{\cos \theta - \sin \theta}

This step is crucial for combining the two fractions into a single term. By manipulating the sign of the second term, we create a common denominator, allowing us to add the numerators.

Step 5: Combine the fractions

Since we now have a common denominator, we can combine the fractions:

cos⁡3θ−sin⁡3θcos⁡θ−sin⁡θ\frac{\cos ^3 \theta - \sin ^3 \theta}{\cos \theta - \sin \theta}

This step is a straightforward application of fraction addition, bringing us closer to simplifying the expression into the desired form.

Step 6: Factor the numerator

The numerator is a difference of cubes, which can be factored using the identity a3−b3=(a−b)(a2+ab+b2)a^3 - b^3 = (a - b)(a^2 + ab + b^2). Applying this to our expression, we get:

cos⁡3θ−sin⁡3θ=(cos⁡θ−sin⁡θ)(cos⁡2θ+cos⁡θsin⁡θ+sin⁡2θ)\cos ^3 \theta - \sin ^3 \theta = (\cos \theta - \sin \theta)(\cos ^2 \theta + \cos \theta \sin \theta + \sin ^2 \theta)

Substituting this back into the LHS, we have:

(cos⁡θ−sin⁡θ)(cos⁡2θ+cos⁡θsin⁡θ+sin⁡2θ)cos⁡θ−sin⁡θ\frac{(\cos \theta - \sin \theta)(\cos ^2 \theta + \cos \theta \sin \theta + \sin ^2 \theta)}{\cos \theta - \sin \theta}

This factorization is a critical step, as it reveals a common factor between the numerator and the denominator, which we can then cancel out.

Step 7: Cancel the common factor

We can now cancel the common factor of (cos⁡θ−sin⁡θ)(\cos \theta - \sin \theta) from the numerator and denominator, provided that cos⁡θ−sin⁡θ≠0\cos \theta - \sin \theta \neq 0:

(cos⁡θ−sin⁡θ)(cos⁡2θ+cos⁡θsin⁡θ+sin⁡2θ)cos⁡θ−sin⁡θ=cos⁡2θ+cos⁡θsin⁡θ+sin⁡2θ\frac{(\cos \theta - \sin \theta)(\cos ^2 \theta + \cos \theta \sin \theta + \sin ^2 \theta)}{\cos \theta - \sin \theta} = \cos ^2 \theta + \cos \theta \sin \theta + \sin ^2 \theta

This simplification brings us significantly closer to the RHS of the identity.

Step 8: Apply the Pythagorean identity

Recall the Pythagorean identity: sin⁡2θ+cos⁡2θ=1\sin ^2 \theta + \cos ^2 \theta = 1. Substituting this into our expression, we get:

cos⁡2θ+cos⁡θsin⁡θ+sin⁡2θ=1+sin⁡θcos⁡θ\cos ^2 \theta + \cos \theta \sin \theta + \sin ^2 \theta = 1 + \sin \theta \cos \theta

This step is the final piece of the puzzle. By applying the Pythagorean identity, we transform the expression into the exact form of the RHS.

Step 9: Conclusion

We have successfully transformed the LHS of the equation into the RHS:

cos⁡2θ1−tan⁡θ+sin⁡3θsin⁡θ−cos⁡θ=1+sin⁡θcos⁡θ\frac{\cos ^2 \theta}{1-\tan \theta}+\frac{\sin ^3 \theta}{\sin \theta-\cos \theta} = 1 + \sin \theta \cos \theta

Thus, we have proven the given trigonometric identity. Each step in this proof has built upon fundamental trigonometric principles and algebraic manipulations, demonstrating the power and elegance of these mathematical tools.

Importance of Practice and Understanding

Proving trigonometric identities requires a solid understanding of the basic trigonometric functions, their definitions, and the fundamental identities. It also involves algebraic skills, such as factoring, simplifying fractions, and manipulating expressions. Practice is essential for mastering these skills.

By working through various examples and exercises, you will become more comfortable with the different techniques used in proving identities. You will also develop a deeper intuition for which identities to apply in different situations. Understanding the underlying concepts is just as important as memorizing formulas. When you grasp the principles behind the identities, you can apply them more effectively and creatively.

Moreover, the ability to prove trigonometric identities is crucial in many areas of mathematics and its applications. It is a fundamental skill for solving trigonometric equations, simplifying complex expressions, and working with calculus and differential equations. Therefore, investing time in understanding and practicing these concepts is a valuable endeavor.

Conclusion

In this article, we have provided a detailed, step-by-step proof of the trigonometric identity:

cos⁡2θ1−tan⁡θ+sin⁡3θsin⁡θ−cos⁡θ=1+sin⁡θcos⁡θ\frac{\cos ^2 \theta}{1-\tan \theta}+\frac{\sin ^3 \theta}{\sin \theta-\cos \theta}=1+\sin \theta \cos \theta

We have demonstrated how to manipulate trigonometric expressions using fundamental identities and algebraic techniques. By breaking down the problem into manageable steps and explaining the reasoning behind each step, we have aimed to make the process clear and accessible.

Mastering trigonometric identities is a crucial skill for anyone studying mathematics, physics, or engineering. It requires a combination of knowledge, practice, and problem-solving ability. We encourage you to continue practicing and exploring the fascinating world of trigonometry. The more you engage with these concepts, the more confident and proficient you will become in applying them.

This journey through the proof highlights not only the specific identity but also the broader importance of trigonometric identities in mathematics. With a firm grasp of these concepts, you'll be well-equipped to tackle more complex problems and appreciate the elegance and power of trigonometry.