Pregunta
Complete the identity. \( \csc x(\sin x+\cos x)= \) ?
Ask by Graham Colon. in the United States
Mar 14,2025
Solución de inteligencia artificial de Upstudy
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\( \csc x(\sin x+\cos x) = 1 + \cot x \)
Solución
Calculate or simplify the expression \( \csc(x)(\sin(x)+\cos(x)) \).
Simplify the expression by following steps:
- step0: Solution:
\(\csc\left(x\right)\left(\sin\left(x\right)+\cos\left(x\right)\right)\)
- step1: Transform the expression:
\(\frac{1}{\sin\left(x\right)}\times \left(\sin\left(x\right)+\cos\left(x\right)\right)\)
- step2: Multiply the terms:
\(\frac{\sin\left(x\right)+\cos\left(x\right)}{\sin\left(x\right)}\)
- step3: Transform the expression:
\(1+\frac{\cos\left(x\right)}{\sin\left(x\right)}\)
- step4: Transform the expression:
\(1+\cot\left(x\right)\)
The completed identity is \( \csc x(\sin x+\cos x) = 1 + \cot x \).
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Simplifique esta solución Beyond the Answer
To complete the identity, we start with the expression given: \(\csc x(\sin x + \cos x)\). Since \(\csc x = \frac{1}{\sin x}\), we can rewrite the expression as: \(\frac{1}{\sin x}(\sin x + \cos x)\). Distributing \(\frac{1}{\sin x}\) gives us: \(\frac{\sin x}{\sin x} + \frac{\cos x}{\sin x} = 1 + \cot x\). So, we have: \(\csc x(\sin x + \cos x) = 1 + \cot x\).
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