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

In this article, we will explore a method to prove the trigonometric identity (tan x sec x frac{cos x}{1 - sin x}). This involves transforming both sides of the equation using basic trigonometric identities to show that they are equivalent.

Introduction

Understanding and verifying trigonometric identities is crucial for advanced mathematics and problem-solving in various fields such as physics and engineering. This article will walk you through the steps to prove this particular identity using fundamental trigonometric properties.

Step-by-Step Proof

Left-Hand Side (LHS)

First, let's start with the left-hand side of the equation: (tan x sec x).

We know that:

(tan x frac{sin x}{cos x}) (sec x frac{1}{cos x})

Thus, we can rewrite the left-hand side as:

[tan x sec x frac{sin x}{cos x} cdot frac{1}{cos x} frac{sin x}{cos^2 x}]

Right-Hand Side (RHS)

Now, let's move on to the right-hand side of the equation: (frac{cos x}{1 - sin x}).

We want to show that both the left-hand side and the right-hand side are equal. To do this, we will manipulate the left-hand side to look like the right-hand side.

Manipulating the Left-Hand Side

Let's represent the left-hand side as:

(frac{sin x}{cos x} cdot frac{1}{cos x} frac{sin x (1 - sin x)}{cos x (1 - sin x)})

This can be written as:

[frac{sin x (1 - sin x)}{cos x (1 - sin x)} frac{1 - sin^2 x}{cos x (1 - sin x)}]

Using the Pythagorean Identity

Recall the Pythagorean identity: (1 - sin^2 x cos^2 x). Substituting this into the equation:

[frac{cos^2 x}{cos x (1 - sin x)} frac{cos x}{1 - sin x}]

This simplifies to:

(frac{cos x}{1 - sin x})

Conclusion

Since we have shown that the left-hand side simplifies to the right-hand side, we can conclude that:

(tan x sec x frac{cos x}{1 - sin x})

This proves the identity. The steps involved in this proof include:

Using the definitions of tangent and secant. Applying the Pythagorean identity. Multiplying by a form of 1 to facilitate simplification.

By understanding these steps, one can verify and prove other trigonometric identities with confidence.