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

Problem 2.538 · hard Beautiful

1. \[ \frac{d}{d x} \left(- \frac{\ln{\left(\cos{\left(x \right)} - 1 \right)}}{2} + \frac{\ln{\left(\cos{\left(x \right)} + 1 \right)}}{2}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = - \frac{d}{d x} \frac{\ln{\left(\cos{\left(x \right)} - 1 \right)}}{2} + \frac{d}{d x} \frac{\ln{\left(\cos{\left(x \right)} + 1 \right)}}{2} \]
   sumApply the sum rule.✓ Proved
3. \[ = - \frac{\frac{d}{d x} \ln{\left(\cos{\left(x \right)} - 1 \right)}}{2} + \frac{\frac{d}{d x} \ln{\left(\cos{\left(x \right)} + 1 \right)}}{2} \]
   constantFactor out the constant coefficients.✓ Proved
4. \[ = \frac{\frac{d}{d x} \left(\cos{\left(x \right)} + 1\right)}{2 \left(\cos{\left(x \right)} + 1\right)} - \frac{\frac{d}{d x} \left(\cos{\left(x \right)} - 1\right)}{2 \left(\cos{\left(x \right)} - 1\right)} \]
   chainApply the chain rule to the logarithmic terms.✓ Proved
5. \[ = - \frac{\sin{\left(x \right)}}{2 \left(\cos{\left(x \right)} + 1\right)} + \frac{\sin{\left(x \right)}}{2 \left(\cos{\left(x \right)} - 1\right)} \]
   derivativeDifferentiate the inner trigonometric functions.✓ Proved
6. \[ = - \frac{\sin{\left(x \right)}}{2 \cos{\left(x \right)} + 2} + \frac{\sin{\left(x \right)}}{2 \cos{\left(x \right)} - 2} \]
   algebraSimplify the signs and coefficients.✓ Proved
7. \[ = \frac{- \left(\cos{\left(x \right)} - 1\right) \sin{\left(x \right)} + \left(\cos{\left(x \right)} + 1\right) \sin{\left(x \right)}}{\left(\cos{\left(x \right)} + 1\right) \left(2 \cos{\left(x \right)} - 2\right)} \]
   algebraFind a common denominator.✓ Proved
8. \[ = \frac{2 \sin{\left(x \right)}}{2 \cos^{2}{\left(x \right)} - 2} \]
   algebra simplifyExpand the numerator and denominator. Combine like terms in the numerator.✓ Proved
9. \[ = \frac{\sin{\left(x \right)}}{\cos^{2}{\left(x \right)} - 1} \]
   simplifyCancel the common factor of 2.✓ Proved
10. \[ = - \frac{\sin{\left(x \right)}}{1 - \cos^{2}{\left(x \right)}} \]
    algebraRewrite the denominator using the identity sin(x)**2 = 1 - cos(x)**2.✓ Proved
11. \[ = - \frac{1}{\sin{\left(x \right)}} \]
    algebra simplifySubstitute the trigonometric identity. Simplify the fraction.✓ Proved
12. \[ = - \csc{\left(x \right)} \]
    simplifyExpress in terms of the cosecant function.✓ Proved