Question
Add or subtract as indicated, and then simplify as much as possible. Leave your answer in terms of sines and cosines only. \( \cos (x)+\frac{1}{\sin (x)}=\square \)
Ask by Mccarthy Smith. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( \cos(x) + \csc(x) \)
Solution
Calculate or simplify the expression \( \cos(x)+\frac{1}{\sin(x)} \).
Simplify the expression by following steps:
- step0: Solution:
\(\cos\left(x\right)+\frac{1}{\sin\left(x\right)}\)
- step1: Transform the expression:
\(\cos\left(x\right)+\csc\left(x\right)\)
The expression \( \cos(x)+\frac{1}{\sin(x)} \) simplifies to \( \cos(x)+\csc(x) \).
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Mind Expander
To combine the terms, first rewrite everything in terms of sine and cosine. The term \(\frac{1}{\sin(x)}\) can be expressed using the identity \(\sec(x) = \frac{1}{\cos(x)}\) as \(\frac{\cos(x)}{\sin(x)\cos(x)}\). So, we have: \[\cos(x) + \frac{1}{\sin(x)} = \cos(x) + \frac{\cos(x)}{\sin(x)\cos(x)} = \frac{\cos^2(x)}{\sin(x)\cos(x)} + \frac{1}{\sin(x)}\] Combine the fractions: \[= \frac{\cos^2(x) + \sin(x)}{\sin(x)\cos(x)}\] Now, simplify this expression as necessary. However, it's best to leave it in this form for clarity in terms of sine and cosine. So, the final answer in terms of sines and cosines is: \[\frac{\cos^2(x) + 1}{\sin(x)}\]
