Whittaker Graham
09/27/2023 · Middle School
Simplify and write the trigonometric expression without any fractions: \( \tan u+\cot u \)
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Tutor-Verified Answer
Step-by-step Solution
Calculate or simplify the expression \( \tan(u)+\cot(u) \).
Simplify the expression by following steps:
- step0: Solution:
\(\tan\left(u\right)+\cot\left(u\right)\)
- step1: Transform the expression:
\(\frac{\sin\left(u\right)}{\cos\left(u\right)}+\cot\left(u\right)\)
- step2: Transform the expression:
\(\frac{\sin\left(u\right)}{\cos\left(u\right)}+\frac{\cos\left(u\right)}{\sin\left(u\right)}\)
- step3: Reduce fractions to a common denominator:
\(\frac{\sin\left(u\right)\sin\left(u\right)}{\cos\left(u\right)\sin\left(u\right)}+\frac{\cos\left(u\right)\cos\left(u\right)}{\sin\left(u\right)\cos\left(u\right)}\)
- step4: Calculate:
\(\frac{\sin\left(u\right)\sin\left(u\right)}{\cos\left(u\right)\sin\left(u\right)}+\frac{\cos\left(u\right)\cos\left(u\right)}{\cos\left(u\right)\sin\left(u\right)}\)
- step5: Calculate:
\(\frac{\sin^{2}\left(u\right)}{\cos\left(u\right)\sin\left(u\right)}+\frac{\cos^{2}\left(u\right)}{\cos\left(u\right)\sin\left(u\right)}\)
- step6: Transform the expression:
\(\frac{\sin^{2}\left(u\right)+\cos^{2}\left(u\right)}{\cos\left(u\right)\sin\left(u\right)}\)
- step7: Rewrite the expression:
\(\frac{1}{\cos\left(u\right)\sin\left(u\right)}\)
- step8: Transform the expression:
\(\frac{1}{\frac{\sin\left(2u\right)}{2}}\)
- step9: Multiply by the reciprocal:
\(1\times \frac{2}{\sin\left(2u\right)}\)
- step10: Calculate:
\(\frac{2}{\sin\left(2u\right)}\)
- step11: Transform the expression:
\(2\csc\left(2u\right)\)
The simplified form of the trigonometric expression \( \tan u + \cot u \) without any fractions is \( 2\csc(2u) \).
Quick Answer
\( 2\csc(2u) \)
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