Simplifying Trigonometric Expressions: sec x - tan x sin x

Understanding the complexities and simplifying expressions involving trigonometric functions is a fundamental skill in mathematics, particularly when dealing with calculus and advanced trigonometry. Let's consider the expression sec x - tan x sin x. This article will guide you through the process of simplifying this expression using basic trigonometric identities.

Step-by-Step Simplification

First, let's recall the basic trigonometric identities:

sec x 1 / cos x tan x sin x / cos x 1 - sin2x cos2x

The given expression is:

[sec x - sin x tan x frac{1}{cos x} - sin x frac{sin x}{cos x}]

We can further simplify this expression as follows:

Convert the expression to a common denominator: Substitute the identities to simplify the expression.

Simplified Expression

Starting from the given expression:

[sec x - sin x tan x frac{1}{cos x} - frac{sin x sin x}{cos x}]

This can be written as:

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

Using the Pythagorean identity, 1 - sin2x cos2x, we can simplify further:

[ frac{cos^2 x}{cos x}]

Finally, we get:

[cos x]

Using Substitution for Simplification

Another method to simplify the expression is by using substitution. Let's define:

[sc sin x cos x]

Then the expression can be simplified as:

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

This demonstrates that, regardless of the method of simplification, the final expression is:

[sec x - tan x sin x cos x]

Conclusion

Understanding and practicing these simplifications can greatly enhance your problem-solving skills in trigonometry. The key is to recognize and apply the basic trigonometric identities effectively. With practice, you can efficiently simplify more complex expressions involving trigonometric functions.

Keywords: trigonometric identities, sec x, sin x, cosine simplification