QuestionJune 5, 2025

33. (1)/(sec(x)+1)+(1)/(sec(x)-1)=2csc(x)cot(x)

33. (1)/(sec(x)+1)+(1)/(sec(x)-1)=2csc(x)cot(x)
33. (1)/(sec(x)+1)+(1)/(sec(x)-1)=2csc(x)cot(x)

Solution
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Answer

The equation is verified as true. Explanation 1. Simplify the Left Side Combine the fractions: \[ \frac{1}{\sec(x) + 1} + \frac{1}{\sec(x) - 1} = \frac{(\sec(x) - 1) + (\sec(x) + 1)}{(\sec(x) + 1)(\sec(x) - 1)} \] Simplify the numerator and denominator: \[ = \frac{2\sec(x)}{\sec^2(x) - 1} \] 2. Use Trigonometric Identity Recognize that \sec^2(x) - 1 = \tan^2(x), so: \[ = \frac{2\sec(x)}{\tan^2(x)} \] 3. Express in Terms of Sine and Cosine Convert to sine and cosine: \[ = \frac{2\frac{1}{\cos(x)}}{\left(\frac{\sin(x)}{\cos(x)}\right)^2} = \frac{2\cos(x)}{\sin^2(x)} \] 4. Simplify to Match Right Side Recognize that \frac{2\cos(x)}{\sin^2(x)} = 2 \csc(x) \cot(x): \[ = 2 \cdot \frac{1}{\sin(x)} \cdot \frac{\cos(x)}{\sin(x)} \]

Explanation

1. Simplify the Left Side<br /> Combine the fractions: <br />\[<br />\frac{1}{\sec(x) + 1} + \frac{1}{\sec(x) - 1} = \frac{(\sec(x) - 1) + (\sec(x) + 1)}{(\sec(x) + 1)(\sec(x) - 1)}<br />\]<br /> Simplify the numerator and denominator:<br />\[<br />= \frac{2\sec(x)}{\sec^2(x) - 1}<br />\]<br /><br />2. Use Trigonometric Identity<br /> Recognize that $\sec^2(x) - 1 = \tan^2(x)$, so:<br />\[<br />= \frac{2\sec(x)}{\tan^2(x)}<br />\]<br /><br />3. Express in Terms of Sine and Cosine<br /> Convert to sine and cosine:<br />\[<br />= \frac{2\frac{1}{\cos(x)}}{\left(\frac{\sin(x)}{\cos(x)}\right)^2} = \frac{2\cos(x)}{\sin^2(x)}<br />\]<br /><br />4. Simplify to Match Right Side<br /> Recognize that $\frac{2\cos(x)}{\sin^2(x)} = 2 \csc(x) \cot(x)$:<br />\[<br />= 2 \cdot \frac{1}{\sin(x)} \cdot \frac{\cos(x)}{\sin(x)}<br />\]
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