Derivative of \( \displaystyle \frac{5 \ln{\left(\cos^{2}{\left(x \right)} - 1 \right)}}{4} - \frac{5 \ln{\left(\cos^{2}{\left(x \right)} \right)}}{4} \)

Problem 2.1411 · hard Beautiful

1. \[ \frac{d}{d x} \left(\frac{5 \ln{\left(\cos^{2}{\left(x \right)} - 1 \right)}}{4} - \frac{5 \ln{\left(\cos^{2}{\left(x \right)} \right)}}{4}\right) \]
   Differentiate the function with respect to x.✓ Proved
2. \[ = \frac{5 \frac{d}{d x} \ln{\left(\cos^{2}{\left(x \right)} - 1 \right)}}{4} - \frac{5 \frac{d}{d x} \ln{\left(\cos^{2}{\left(x \right)} \right)}}{4} \]
   constant-multipleApply the constant multiple rule to each term.✓ Proved
3. \[ = - \frac{5 \frac{d}{d x} \cos^{2}{\left(x \right)}}{4 \cos^{2}{\left(x \right)}} + \frac{5 \frac{d}{d x} \left(\cos^{2}{\left(x \right)} - 1\right)}{4 \left(\cos^{2}{\left(x \right)} - 1\right)} \]
   chainApply the chain rule to the logarithmic terms.✓ Proved
4. \[ = - \frac{5 \frac{d}{d x} \cos{\left(x \right)}}{2 \cos{\left(x \right)}} + \frac{5 \cos{\left(x \right)} \frac{d}{d x} \cos{\left(x \right)}}{2 \left(\cos^{2}{\left(x \right)} - 1\right)} \]
   chainApply the chain rule to the squared terms.✓ Proved
5. \[ = \frac{5 \sin{\left(x \right)}}{2 \cos{\left(x \right)}} - \frac{5 \sin{\left(x \right)} \cos{\left(x \right)}}{2 \left(\cos^{2}{\left(x \right)} - 1\right)} \]
   trig algebra algebra constant-multipleDifferentiate cos(x). Simplify the products and quotients. Simplify the second term by canceling cos(x). Simplify the coefficients.✓ Proved
6. \[ = \frac{5 \tan{\left(x \right)}}{2} + \frac{5 \cos{\left(x \right)}}{2 \sin{\left(x \right)}} \]
   algebra algebraUse the identity cos(x)**2 - 1 = -sin(x)**2. Simplify the first term by canceling sin(x).✓ Proved
7. \[ = \frac{5 \tan{\left(x \right)}}{2} + \frac{5}{2 \tan{\left(x \right)}} \]
   algebraRewrite cos(x)/sin(x) as 1/tan(x).✓ Proved
8. \[ = \frac{5 \tan{\left(x \right)}}{2} + \frac{5 \cot{\left(x \right)}}{2} \]
   algebraFactor out the common term.✓ Proved
9. \[ = \frac{5 \sin{\left(x \right)}}{2 \cos{\left(x \right)}} + \frac{5 \cos{\left(x \right)}}{2 \sin{\left(x \right)}} \]
   algebraRewrite cot(x) as cos(x)/sin(x).✓ Proved
10. \[ = \frac{\frac{5 \sin^{2}{\left(x \right)}}{2} + \frac{5 \cos^{2}{\left(x \right)}}{2}}{\sin{\left(x \right)} \cos{\left(x \right)}} \]
    algebraCombine the terms under a common denominator.✓ Proved
11. \[ = \frac{5}{2 \sin{\left(x \right)} \cos{\left(x \right)}} \]
    simplify algebraUse the Pythagorean identity sin(x)**2 + cos(x)**2 = 1. Distribute the 5/2.✓ Proved
12. \[ = \frac{5}{\sin{\left(2 x \right)}} \]
    trigUse the double angle identity 2*sin(x)*cos(x) = sin(2*x).✓ Proved
13. \[ = 5 \csc{\left(2 x \right)} \]
    simplifyRewrite 1/sin(2*x) as csc(2*x).✓ Proved