The given mathematical expression is:

$\sec x - \tan x \sin x$

Step 1: Express in Terms of Sine and Cosine

We use the trigonometric identities:

• $\sec x = \frac{1}{\cos x}$
• $\tan x = \frac{\sin x}{\cos x}$

Rewriting the given expression:

$\frac{1}{\cos x} - \left( \frac{\sin x}{\cos x} \cdot \sin x \right)$

Step 2: Simplify the Expression

$\frac{1}{\cos x} - \frac{\sin^2 x}{\cos x}$

Since both terms have a common denominator $\cos x$, we combine them:

$\frac{1 - \sin^2 x}{\cos x}$

Step 3: Apply the Pythagorean Identity

From the identity:

$\sin^2 x + \cos^2 x = 1$

We replace $1 - \sin^2 x$ with $\cos^2 x$:

$\frac{\cos^2 x}{\cos x}$

Step 4: Final Simplification

$\cos x$

Final Answer:

$\cos x$

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Would you like a detailed explanation of any step?

Related Questions:

1. What are the fundamental trigonometric identities?
2. How do you derive $\tan x$ and $\sec x$ using sine and cosine?
3. Why is the identity $\sin^2 x + \cos^2 x = 1$ important in trigonometry?
4. How can we simplify expressions using trigonometric identities?
5. What are some real-world applications of trigonometric identities?

Tip: Recognizing common trigonometric identities can help simplify complex expressions quickly.