Proving the Trigonometric Identity: tan(x/2) (1 - cos(x)) / sin(x)

The identity tan(x/2) (1 - cos(x)) / sin(x) is a fundamental relationship in trigonometry that can be useful in simplifying complex trigonometric expressions and solving trigonometric equations. In this article, we will explore how this identity can be proven using double angle identities and basic trigonometric properties.

Understanding the Double Angle Identities

Before we prove the identity, let’s briefly review some key double angle identities:

Double Angle Identity for Sine:

sin(2y) 2sin(y)cos(y)

Double Angle Identity for Cosine:

cos(2y) 2cos^2(y) - 1 and cos(2y) 1 - 2sin^2(y)

Proving the Identity: tan(x/2) (1 - cos(x)) / sin(x)

Let's start by expressing the given identity in a form that we can prove. We aim to show that:

tan(x/2) (1 - cos(x)) / sin(x)

Step 1: Expressing sin(x) and cos(x) using Double Angle Identities

We know from the double angle identities:

sin(x) 2sin(x/2)cos(x/2)

Rearranging to solve for sin(x/2) and cos(x/2), we get:

sin(x/2) sin(x) / (2cos(x/2))

cos(x/2) cos(x) / (2sin(x/2)), but we will use the primary expression for now.

Using the double angle identity for cosine:

cos(x) 1 - 2sin^2(x/2)

Step 2: Simplifying the Right-Hand Side (RHS)

Let's simplify the right-hand side of the identity first:

RHS (1 - cos(x)) / sin(x)

Substitute the expression for cos(x) obtained from the double angle identity:

RHS (1 - (1 - 2sin^2(x/2))) / sin(x)

RHS (2sin^2(x/2)) / sin(x)

Step 3: Further Simplification

Now, we can factor out sin(x/2) from the numerator and rewrite the expression:

RHS 2sin^2(x/2) / (2sin(x/2)cos(x/2))

RHS sin(x/2) / cos(x/2)

This gives us:

RHS tan(x/2)

Conclusion

We have shown that:

tan(x/2) (1 - cos(x)) / sin(x)

This proves the given identity. The process involves expressing sin(x) and cos(x) using the double angle identities, simplifying the right-hand side, and then showing that it equals the left-hand side.

Applications of the Identity

This identity is particularly useful in various areas of trigonometry, including integration, solving trigonometric equations, and simplifying complicated trigonometric expressions. For instance, it can be used to integrate expressions involving tan(x/2) or simplify complex trigonometric equations.

Practice Problems

1. Prove that tan(x/2) (1 - cos(x)) / sin(x) using alternative methods.

2. Use the identity to simplify the expression sin^2(x) cos^2(x) - 2sin^2(x/2).

3. Solve the equation tan(2x) 2tan(x) / (1 - tan^2(x)) using the identity and double angle formulas.

Understanding and mastering such identities is crucial for any student of calculus and trigonometry, as it enhances problem-solving skills and provides a solid foundation for advanced mathematical concepts.