Sure! Let's prove the identity step by step:
[tex]\[
\sec(x) - \sin(x) \tan(x) = \cos(x)
\][/tex]
Step 1: Understand the Trigonometric Functions
We know the following trigonometric identities:
[tex]\[
\sec(x) = \frac{1}{\cos(x)}, \quad \tan(x) = \frac{\sin(x)}{\cos(x)}
\][/tex]
Step 2: Rewrite the Left Side of the Equation
Let's start by rewriting the left-hand side of the given identity using the definitions of [tex]\(\sec(x)\)[/tex] and [tex]\(\tan(x)\)[/tex]:
[tex]\[
\sec(x) - \sin(x) \tan(x) = \frac{1}{\cos(x)} - \sin(x) \cdot \frac{\sin(x)}{\cos(x)}
\][/tex]
Step 3: Combine the Terms over a Common Denominator
Since both terms on the left-hand side have a common denominator, we can combine them:
[tex]\[
\frac{1}{\cos(x)} - \sin(x) \cdot \frac{\sin(x)}{\cos(x)} = \frac{1 - \sin^2(x)}{\cos(x)}
\][/tex]
Step 4: Use the Pythagorean Identity
Recall the Pythagorean identity:
[tex]\[
\sin^2(x) + \cos^2(x) = 1
\][/tex]
We can use this identity to replace [tex]\(1 - \sin^2(x)\)[/tex] with [tex]\(\cos^2(x)\)[/tex]:
[tex]\[
\frac{1 - \sin^2(x)}{\cos(x)} = \frac{\cos^2(x)}{\cos(x)}
\][/tex]
Step 5: Simplify the Expression
Now we simplify the fraction:
[tex]\[
\frac{\cos^2(x)}{\cos(x)} = \cos(x)
\][/tex]
Step 6: Conclusion
Hence, we have shown that:
[tex]\[
\sec(x) - \sin(x) \tan(x) = \cos(x)
\][/tex]
Thus, the identity is proved!
