Home›Calculus 1›Derivatives of trigonometric functions›Problem 2.970

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

Problem 2.970 · hard Beautiful

Differentiate \( \displaystyle f(x) = \ln{\left(\cos{\left(x \right)} - 1 \right)} - \ln{\left(\cos{\left(x \right)} + 1 \right)} \).

\[ \frac{d}{d x} \left(\ln{\left(\cos{\left(x \right)} - 1 \right)} - \ln{\left(\cos{\left(x \right)} + 1 \right)}\right) \]
Start with the derivative of the function.✓ Proved
\[ = \frac{d}{d x} \ln{\left(\cos{\left(x \right)} - 1 \right)} - \frac{d}{d x} \ln{\left(\cos{\left(x \right)} + 1 \right)} \]
sumApply the sum rule for subtraction.✓ Proved
\[ = \frac{d}{d x} \ln{\left(\cos{\left(x \right)} - 1 \right)} + \frac{d}{d x} \left(- \ln{\left(\cos{\left(x \right)} + 1 \right)}\right) \]
algebraRewrite the subtraction as addition of a negative.✓ Proved
\[ = \frac{d}{d x} \ln{\left(\cos{\left(x \right)} - 1 \right)} - \frac{d}{d x} \ln{\left(\cos{\left(x \right)} + 1 \right)} \]
constantThe constant factor -1 is moved out.✓ Proved
\[ = - \frac{\frac{d}{d x} \left(\cos{\left(x \right)} + 1\right)}{\cos{\left(x \right)} + 1} + \frac{\frac{d}{d x} \left(\cos{\left(x \right)} - 1\right)}{\cos{\left(x \right)} - 1} \]
chainApply the chain rule to both logarithmic terms.✓ Proved
\[ = - \frac{\frac{d}{d x} \cos{\left(x \right)}}{\cos{\left(x \right)} + 1} + \frac{\frac{d}{d x} \cos{\left(x \right)}}{\cos{\left(x \right)} - 1} \]
derivativeDifferentiate the inner functions (cos(x) - 1) and (cos(x) + 1).✓ Proved
\[ = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)} + 1} - \frac{\sin{\left(x \right)}}{\cos{\left(x \right)} - 1} \]
trig algebraThe derivative of cos(x) is -sin(x). Simplify the signs.✓ Proved
\[ = \left(\frac{1}{\cos{\left(x \right)} + 1} - \frac{1}{\cos{\left(x \right)} - 1}\right) \sin{\left(x \right)} \]
algebraFactor out sin(x).✓ Proved
\[ = - \frac{2 \sin{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right) \left(\cos{\left(x \right)} + 1\right)} \]
algebraCombine the fractions using a common denominator.✓ Proved
\[ = - \frac{2 \sin{\left(x \right)}}{\cos^{2}{\left(x \right)} - 1} \]
algebra algebraSimplify the numerator and denominator. Distribute sin(x).✓ Proved
\[ = \frac{2}{\sin{\left(x \right)}} \]
algebra simplifyUse the identity cos(x)**2 - 1 = -sin(x)**2. Simplify the expression by canceling -sin(x).✓ Proved
\[ = 2 \csc{\left(x \right)} \]
simplifyRewrite 1/sin(x) as csc(x).✓ Proved

Answer \( \frac{2}{\sin{\left(x \right)}} \)

Mind the domain. The answer is also defined at points where f(x) is not. Substituting there gives a number that is not a slope of f.

Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.

The full receipt

Line	Status	Checked by	Detail
1	✓ Proved	sympy 1.14.0	line 1 is the problem as stated
2	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
3	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
4	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments
5	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 log is undefined for non-positive arguments undefined where cos(x) + 1 = 0 undefined where cos(x) - 1 = 0
6	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(x) + 1 = 0 undefined where cos(x) - 1 = 0
7	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(x) + 1 = 0 undefined where cos(x) - 1 = 0
8	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(x) + 1 = 0 undefined where cos(x) - 1 = 0
9	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(x) + 1 = 0 undefined where cos(x) - 1 = 0
10	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(x) + 1 = 0 undefined where cos(x) - 1 = 0
11	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(x) + 1 = 0 undefined where cos(x) - 1 = 0 undefined where cos(x)**2 - 1 = 0
12	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(x)**2 - 1 = 0
13	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where cos(x)**2 - 1 = 0 undefined where sin(x) = 0
14	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(x) = 0
15	✓ Proved	sympy 1.14.0	simplify(a - b) reduced to 0 undefined where sin(x) = 0 csc has poles at multiples of pi
answer	✓ Proved	sympy 1.14.0	final line against the stated answer: simplify(a - b) reduced to 0 undefined where sin(x) = 0
answer, a second way	✓ Proved	sympy 1.14.0	SymPy differentiated f directly and got the stated answer

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — Step 5 applies the chain rule to both terms simultaneously, violating the one-rule-per-step constraint. Additionally, Step 4 is logically flawed: it claims to move a constant factor out (which would require the constant-multiple rule) but the expression remains a subtraction, effectively doing nothing or mislabeling the operation.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-09-27 — Step 5 applies the chain rule to both terms simultaneously, violating the one-rule-per-step constraint. Additionally, Step 4 is logically flawed: it claims to move a constant factor out (which would require the constant-multiple rule) but the expression remains a subtraction, effectively doing nothing or mislabeling the operation.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-27
gpt-oss:20b: pass 2026-09-27

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by gemma4:26b, checked 2026-09-27 with SymPy 1.14.0.