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Complete the identity. \( \tan x(\cot x-\cos x)=? \)

Ask by Owen Fowler. in the United States
Mar 14,2025

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Tutor-Verified Answer

Answer

\( \tan x(\cot x-\cos x) = 1 - \sin x \)

Solution

Calculate or simplify the expression \( \tan(x)(\cot(x)-\cos(x)) \). Simplify the expression by following steps: - step0: Solution: \(\tan\left(x\right)\left(\cot\left(x\right)-\cos\left(x\right)\right)\) - step1: Transform the expression: \(\frac{\sin\left(x\right)}{\cos\left(x\right)}\times \left(\cot\left(x\right)-\cos\left(x\right)\right)\) - step2: Transform the expression: \(\frac{\sin\left(x\right)}{\cos\left(x\right)}\times \frac{\cos\left(x\right)-\cos\left(x\right)\sin\left(x\right)}{\sin\left(x\right)}\) - step3: Reduce the fraction: \(\frac{1}{\cos\left(x\right)}\times \left(\cos\left(x\right)-\cos\left(x\right)\sin\left(x\right)\right)\) - step4: Rewrite the expression: \(\frac{1}{\cos\left(x\right)}\times \cos\left(x\right)\left(1-\sin\left(x\right)\right)\) - step5: Reduce the fraction: \(1\times \left(1-\sin\left(x\right)\right)\) - step6: Multiply the terms: \(1-\sin\left(x\right)\) The completed identity is \( \tan x(\cot x-\cos x) = 1 - \sin x \).

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Simplify this solution

Mind Expander

To complete the identity, let's simplify the expression step-by-step. Starting with the expression: \[ \tan x(\cot x - \cos x) \] We know: \[ \tan x = \frac{\sin x}{\cos x} \quad \text{and} \quad \cot x = \frac{\cos x}{\sin x} \] Substituting these definitions in, we have: \[ \tan x(\cot x - \cos x) = \frac{\sin x}{\cos x} \left(\frac{\cos x}{\sin x} - \cos x\right) \] Now simplify the term inside the parentheses: \[ \frac{\cos x}{\sin x} - \cos x = \frac{\cos x - \cos x \sin x}{\sin x} = \frac{\cos x(1 - \sin x)}{\sin x} \] Now substitute this back into the original expression: \[ \tan x \cdot \frac{\cos x(1 - \sin x)}{\sin x} = \frac{\sin x}{\cos x} \cdot \frac{\cos x(1 - \sin x)}{\sin x} \] The \(\sin x\) in the numerator and denominator cancels out, leaving us with: \[ \frac{\cancel{\sin x}}{\cos x} \cdot \frac{\cos x(1 - \sin x)}{\cancel{\sin x}} = 1 - \sin x \] Thus, the completed identity is: \[ \tan x(\cot x - \cos x) = 1 - \sin x \]

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