Proving the Trigonometric Identity: tan(x) (frac{sin^2(x)}{1-cos(x)} - frac{1}{sin(x)})
This tutorial will guide you through the process of proving a challenging trigonometric identity: tan(x) (frac{sin^2(x)}{1-cos(x)} - frac{1}{sin(x)}). The steps will involve rationalizing expressions and simplifying trigonometric functions to prove the equality. This method is crucial for understanding the relationships between sine, cosine, and tangent functions, and it will be demonstrated through detailed algebraic manipulations.
Step-by-Step Proof
Left Hand Side Transformation
First, let's look at the left-hand side of the equation:
(frac{sin^2(x)}{1-cos(x)} - frac{1}{sin(x)})
We will use algebraic identities to simplify each term. Let's rationalize the terms by expressing them in a form that will help us see the relationship more clearly.
Term 1: (frac{sin^2(x)}{1-cos(x)})
We can rewrite (frac{sin^2(x)}{1-cos(x)}) as follows:
(frac{sin^2(x)}{1-cos(x)} frac{1 - cos^2(x)}{1-cos(x)})
Using the identity (1 - cos^2(x) sin^2(x)), we get:
(frac{1 - cos^2(x)}{1-cos(x)} frac{(1-cos(x))(1 cos(x))}{1-cos(x)})
Which simplifies to:
(1 cos(x))
Term 2: (frac{1}{sin(x)})
The second term is simply:
(frac{1}{sin(x)})
Combining the Terms
Now, we combine the simplified terms:
(frac{sin^2(x)}{1-cos(x)} - frac{1}{sin(x)} (1 cos(x)) - frac{1}{sin(x)})
Right Hand Side Simplification
Let's simplify the right-hand side of the equation:
(frac{1}{tan(x)} cot(x) frac{cos(x)}{sin(x)} frac{cos(x) - cos^2(x)}{sin(x) - cos(x)})
This can be further simplified as:
(frac{cos(x) - cos^2(x)}{sin(x) - cos(x)} frac{cos(x) - 1 - sin^2(x)}{sin(x) - cos(x)})
Since (1 sin^2(x) cos^2(x)), we can rewrite the numerator:
(cos(x) - 1 - sin^2(x) cos(x) - cos^2(x))
Thus, the right-hand side becomes:
(frac{cos(x) - cos^2(x)}{sin(x) - cos(x)} frac{cos(x)(1 - cos(x))}{sin(x) - cos(x)} frac{cos(x)}{sin(x)})
Which simplifies to:
(frac{cos(x)}{sin(x)} frac{1}{tan(x)})
Conclusion
Through these detailed steps, we have successfully proven that:
tan(x) (frac{sin^2(x)}{1-cos(x)} - frac{1}{sin(x)})
This method of algebraic manipulation and simplification is a fundamental technique in trigonometry. Understanding such identities is essential for solving complex trigonometric equations and problem-solving in mathematics. Practice with similar problems and identities will reinforce your skills in trigonometric manipulations.