To simplify the expression [tex]\(\frac{\sin x - \cos x}{\sin x}\)[/tex], let's break it down step by step.
1. Rewrite the Expression:
[tex]\[\frac{\sin x - \cos x}{\sin x}\][/tex]
2. Separate the Terms in the Numerator:
Rewrite the fraction as the sum of two fractions:
[tex]\[\frac{\sin x}{\sin x} - \frac{\cos x}{\sin x}\][/tex]
3. Simplify Each Term:
- The first term simplifies to 1 because [tex]\(\frac{\sin x}{\sin x} = 1\)[/tex].
- The second term can be rewritten using the reciprocal of [tex]\(\sin x\)[/tex], which is [tex]\(\csc x\)[/tex]:
[tex]\[\frac{\cos x}{\sin x} = \cot x\][/tex]
Therefore, the second term simplifies to [tex]\(\cot x\)[/tex].
4. Combine the Simplified Terms:
Now, combine the terms we obtained:
[tex]\[1 - \cot x\][/tex]
5. Further Simplification:
Recall that [tex]\(\cot x\)[/tex] can also be expressed as [tex]\(\frac{1}{\tan x}\)[/tex]. Hence:
[tex]\[1 - \cot x = 1 - \frac{1}{\tan x}\][/tex]
So, the simplified form of the given expression [tex]\(\frac{\sin x - \cos x}{\sin x}\)[/tex] is:
[tex]\[\boxed{1 - \frac{1}{\tan x}}\][/tex]
