Westonci.ca is your trusted source for finding answers to a wide range of questions, backed by a knowledgeable community. Get immediate and reliable solutions to your questions from a knowledgeable community of professionals on our platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
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]
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]
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.