How do sine and cosine give tangent and cotangent derivatives?
You will be able to: Derive the tangent and cotangent rules using identities and the quotient rule.
BC foundation: Unit 2 shares its derivative foundations with AB. Connect each rule to rates, graphs and its domain conditions. Use the written challenges to explain your reasoning. Chain-rule compositions, implicit and inverse differentiation, and advanced applications come later.
How do sine and cosine give tangent and cotangent derivatives?
A direction angle can be described by the ratio of its vertical and horizontal components. Differentiating that ratio connects new trigonometric rules to familiar sine and cosine rates.
A useful starting point: When does simplifying help, and what stays excluded? →
Words and symbols before equations
- tan x
- sin x/cos x, requiring cos x≠0.
- cot x
- cos x/sin x, requiring sin x≠0.
- sec x
- 1/cos x.
- csc x
- 1/sin x.
- Pythagorean identity
- sin²x+cos²x=1.
What this picture assumes
Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. Radian input on 0<x<π/2, where both functions exist. The finite plotted branch does not include or join across asymptotes. Full restrictions are stated in the lesson.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- tan at radian input 0.8: value=1.02964, derivative=2.06016. Blue: function; orange: tangent; second graph: derivative. Curves outside the finite vertical scale are clipped, not capped. Use radians and retain denominator exclusions.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Using radians, differentiate tan x=sin x/cos x. Its numerator is cos²x−sin x(−sin x)=cos²x+sin²x=1, so the derivative is 1/cos²x=sec²x.
For cot x=cos x/sin x the numerator is (−sin x)sin x−cos²x=−1. Thus the derivative is −1/sin²x=−csc²x.
Both formulas retain the original denominator restrictions. There is no derivative at the functions’ vertical asymptotes.
The signs differ: tangent has positive slope on each allowed branch, whereas cotangent has negative slope. These names refer to trig functions; a tangent line is a different use of the word tangent.
A worked example, step by step
Find the derivatives of tan x and cot x at x=π/4.
- At π/4, sine and cosine both equal sqrt(2)/2.
- For tangent, sec²x=1/(1/2)=2.
- For cotangent, −csc²x=−1/(1/2)=−2.
- Both are defined there; all inputs use radians.
The cotangent derivative is negative. Neither rule grants a derivative at an excluded asymptote.
Is tan x differentiable at π/2?
Compare with an explanation
No. cos(π/2)=0, so tan itself is undefined there.
Predict. Change one thing. Explain.
Compare tangent and cotangent on the same valid radian interval. Move the input toward an edge and explain the increasingly steep slopes using the squared denominator.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
tan at radian input 0.8: value=1.02964, derivative=2.06016. Blue: function; orange: tangent; second graph: derivative. Curves outside the finite vertical scale are clipped, not capped. Use radians and retain denominator exclusions.
Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. Radian input on 0<x<π/2, where both functions exist. The finite plotted branch does not include or join across asymptotes. Full restrictions are stated in the lesson.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using a difference quotient, tangent slope, or derivative rule with its domain conditions. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionDerive the derivative of cot x from cos x/sin x, including its domain condition.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Apply the quotient rule with u=cos x and v=sin x.
- 1 point: The numerator is −sin²x−cos²x=−1.
- 1 point: Divide by sin²x to obtain −csc²x.
- 1 point: Require sin x≠0, with radian input.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1Tangent derivative?
sec²x where defined.
RECALL 2Cotangent derivative?
−csc²x where defined.
RECALL 3Which identity simplifies the numerators?
sin²x+cos²x=1.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do sine and cosine give tangent and cotangent derivatives?
- (tan x)′=sec²x where cos x≠0.
- (cot x)′=−csc²x where sin x≠0.
Remember: The cotangent derivative is negative. Neither rule grants a derivative at an excluded asymptote.
Conditions: Original equation-driven model; readouts are rounded. Graphs have labeled linear scales and finite sampled windows; algebra supplies exact conclusions. Radian input on 0<x<π/2, where both functions exist. The finite plotted branch does not include or join across asymptotes. Full restrictions are stated in the lesson.
Refresh Kid · AP Calculus BC Unit 2 · Objectives FUN-3.B · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 2.10, FUN-3.B. CED effective Fall 2020, with Fall 2026 clarifications, checked September 17, 2026. Unit 2 has 10 official topics. Focused lesson titles and questions are original Refresh Kid teaching material.
Derivative definitions require finite two-sided limits at interior inputs. Graphs and finite tables supply evidence and estimates, not universal proofs. Continuity is necessary but insufficient for differentiability. Rules retain their original domain restrictions, and trigonometric inputs use radians. Chain-rule compositions, implicit differentiation, inverse-function differentiation and L’Hôpital’s rule are deferred to later units. Piecewise joins are checked for continuity before their side slopes are compared.
The Organic Chemistry Tutor video titles, creators and descriptions were checked; full videos were not reviewed. Khan Academy’s unit destination was checked; its JavaScript lesson content was not fully readable by the research tool. OpenStax Sections 3.1, 3.3 and 3.5 were consulted for conceptual cross-checking. No creator scripts, questions, diagrams or artwork were copied. Refresh Kid is not endorsed by these providers.
GitHub’s 3D website collection and its camera-control example informed optional spatial inspection. Graph geometry is generated from the stated original equations; self-hosted Three.js retains its MIT license. The optional 3D model uses separate planes for f and f′. Depth distinguishes panels, not an additional variable; camera rotation does not change their values. Use the complete labeled 2D graphs and text to read precise coordinates and slopes.
Independent teacher review and observation of students remain pending. Checks do not certify scientific accuracy, accessibility or learning effectiveness. This is a review edition.
Optional official resource: Released AP Calculus BC questions and scoring guides. The archive spans multiple units; no entire exam question is assigned to this lesson.
Learn → Explore → Practice → Review is informed by the IES learning guide; this exact implementation has not been evaluated with learners.
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