QuestionJuly 14, 2025

5. Factor to write each in a simpler form. (cotTheta +tanTheta )/(sec(-Theta ))

5. Factor to write each in a simpler form. (cotTheta +tanTheta )/(sec(-Theta ))
5. Factor to write each in a simpler form. (cotTheta +tanTheta )/(sec(-Theta ))

Solution
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Answer

\csc\Theta Explanation 1. Simplify sec(-\Theta) Use the identity sec(-\Theta) = sec(\Theta). 2. Simplify cot\Theta + tan\Theta Use identities: cot\Theta = \frac{cos\Theta}{sin\Theta} and tan\Theta = \frac{sin\Theta}{cos\Theta}. Combine: \frac{cos\Theta}{sin\Theta} + \frac{sin\Theta}{cos\Theta} = \frac{cos^2\Theta + sin^2\Theta}{sin\Theta \cdot cos\Theta}. 3. Apply Pythagorean Identity cos^2\Theta + sin^2\Theta = 1, so \frac{1}{sin\Theta \cdot cos\Theta}. 4. Simplify the Expression Divide by sec(\Theta): \frac{1}{sin\Theta \cdot cos\Theta \cdot \frac{1}{cos\Theta}} = \frac{1}{sin\Theta}.

Explanation

1. Simplify $sec(-\Theta)$<br /> Use the identity $sec(-\Theta) = sec(\Theta)$.<br />2. Simplify $cot\Theta + tan\Theta$<br /> Use identities: $cot\Theta = \frac{cos\Theta}{sin\Theta}$ and $tan\Theta = \frac{sin\Theta}{cos\Theta}$. Combine: $\frac{cos\Theta}{sin\Theta} + \frac{sin\Theta}{cos\Theta} = \frac{cos^2\Theta + sin^2\Theta}{sin\Theta \cdot cos\Theta}$.<br />3. Apply Pythagorean Identity<br /> $cos^2\Theta + sin^2\Theta = 1$, so $\frac{1}{sin\Theta \cdot cos\Theta}$.<br />4. Simplify the Expression<br /> Divide by $sec(\Theta)$: $\frac{1}{sin\Theta \cdot cos\Theta \cdot \frac{1}{cos\Theta}} = \frac{1}{sin\Theta}$.
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