Derivative of \( \displaystyle - \frac{\ln{\left(\sin{\left(x \right)} - 1 \right)}}{2} + \frac{\ln{\left(\sin{\left(x \right)} + 1 \right)}}{2} \)

Problem 2.217 · hard Beautiful

1. \[ \frac{d}{d x} \left(- \frac{\ln{\left(\sin{\left(x \right)} - 1 \right)}}{2} + \frac{\ln{\left(\sin{\left(x \right)} + 1 \right)}}{2}\right) \]
   sum constant-multiple algebraApply the sum rule to differentiate each term separately. Distribute the division by 2.✓ Proved
2. \[ = \frac{d}{d x} \left(- \frac{\ln{\left(\sin{\left(x \right)} - 1 \right)}}{2}\right) + \frac{d}{d x} \frac{\ln{\left(\sin{\left(x \right)} + 1 \right)}}{2} \]
   sumSplit the derivative into two parts.✓ Proved
3. \[ = - \frac{\frac{d}{d x} \ln{\left(\sin{\left(x \right)} - 1 \right)}}{2} + \frac{\frac{d}{d x} \ln{\left(\sin{\left(x \right)} + 1 \right)}}{2} \]
   constant-multiple✓ Proved
4. \[ = \frac{\frac{d}{d x} \left(\sin{\left(x \right)} + 1\right)}{2 \left(\sin{\left(x \right)} + 1\right)} - \frac{\frac{d}{d x} \left(\sin{\left(x \right)} - 1\right)}{2 \left(\sin{\left(x \right)} - 1\right)} \]
   chainApply the chain rule to both logarithmic terms.✓ Proved
5. \[ = \frac{\cos{\left(x \right)}}{2 \left(\sin{\left(x \right)} + 1\right)} - \frac{\cos{\left(x \right)}}{2 \left(\sin{\left(x \right)} - 1\right)} \]
   derivativeDifferentiate the inner functions sin(x) - 1 and sin(x) + 1.✓ Proved
6. \[ = \frac{\cos{\left(x \right)}}{2 \sin{\left(x \right)} + 2} - \frac{\cos{\left(x \right)}}{2 \sin{\left(x \right)} - 2} \]
   algebra algebraSimplify the product of the fractions. Rearrange the terms.✓ Proved
7. \[ = \frac{\left(\sin{\left(x \right)} - 1\right) \cos{\left(x \right)} - \left(\sin{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\left(\sin{\left(x \right)} - 1\right) \left(2 \sin{\left(x \right)} + 2\right)} \]
   algebraFind a common denominator.✓ Proved
8. \[ = - \frac{2 \cos{\left(x \right)}}{2 \sin^{2}{\left(x \right)} - 2} \]
   algebra algebra algebraExpand the numerator and denominator. Combine like terms in the numerator.✓ Proved
9. \[ = - \frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)} - 1} \]
   simplifyCancel the common factor of 2.✓ Proved
10. \[ = \frac{\cos{\left(x \right)}}{1 - \sin^{2}{\left(x \right)}} \]
    algebraDistribute the negative sign into the denominator.✓ Proved
11. \[ = \frac{1}{\cos{\left(x \right)}} \]
    rewrite simplifyUse the trigonometric identity 1 - sin(x)**2 = cos(x)**2. Simplify the fraction.✓ Proved
12. \[ = \sec{\left(x \right)} \]
    rewriteRewrite 1/cos(x) as sec(x).✓ Proved