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

Problem 2.945 · hard Beautiful

1. \[ \frac{d}{d x} \left(\frac{\ln{\left(\cos^{2}{\left(x \right)} - 1 \right)}}{4} - \frac{\ln{\left(\cos^{2}{\left(x \right)} \right)}}{4}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \frac{\ln{\left(\cos^{2}{\left(x \right)} - 1 \right)}}{4} - \frac{d}{d x} \frac{\ln{\left(\cos^{2}{\left(x \right)} \right)}}{4} \]
   sumApply the difference rule.✓ Proved
3. \[ = \frac{\frac{d}{d x} \ln{\left(\cos^{2}{\left(x \right)} - 1 \right)}}{4} - \frac{\frac{d}{d x} \ln{\left(\cos^{2}{\left(x \right)} \right)}}{4} \]
   constantFactor out the constant 1/4.✓ Proved
4. \[ = - \frac{\frac{d}{d x} \cos^{2}{\left(x \right)}}{4 \cos^{2}{\left(x \right)}} + \frac{\frac{d}{d x} \left(\cos^{2}{\left(x \right)} - 1\right)}{4 \left(\cos^{2}{\left(x \right)} - 1\right)} \]
   chainApply the chain rule to both logarithmic terms.✓ Proved
5. \[ = - \frac{\frac{d}{d x} \cos{\left(x \right)}}{2 \cos{\left(x \right)}} + \frac{\cos{\left(x \right)} \frac{d}{d x} \cos{\left(x \right)}}{2 \left(\cos^{2}{\left(x \right)} - 1\right)} \]
   chainApply the chain rule to the inner squared terms.✓ Proved
6. \[ = \frac{\sin{\left(x \right)}}{2 \cos{\left(x \right)}} - \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{2 \left(\cos^{2}{\left(x \right)} - 1\right)} \]
   trig algebra algebraDifferentiate cos(x). Simplify the products in the numerators. Distribute the negative sign and simplify coefficients.✓ Proved
7. \[ = \frac{\sin{\left(x \right)}}{2 \cos{\left(x \right)}} + \frac{\cos{\left(x \right)}}{2 \sin{\left(x \right)}} \]
   algebra algebra algebraUse the identity cos(x)**2 - 1 = -sin(x)**2. Simplify the signs. Cancel the common sin(x) and cos(x) terms.✓ Proved
8. \[ = \frac{\tan{\left(x \right)}}{2} + \frac{\cot{\left(x \right)}}{2} \]
   rewrite algebraRewrite the fractions using cotangent and tangent. Factor out 1/2.✓ Proved
9. \[ = \frac{1}{\sin{\left(2 x \right)}} \]
   simplifyUse the identity cot(x) + tan(x) = 2/sin(2x) to simplify.≈ Checked numerically