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Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions

Derive reciprocal-trigonometric rules from identities and track signs, radians, and domains.

This is shared AB/BC instruction. Open the AB edition of Topic 2.10. Your BC practice and progress stay in this book.

College Board Topic 2.10 · CED p. 64 (PDF p. 69) · CED effective Fall 2020; project course outline 2026–27.

Use trig identities to connect new derivative rules with the ones already known. Write tan x=sin x/cos x and cot x=cos x/sin x. The quotient rule and sin²x+cos²x=1 give (tan x)′=sec²x and (cot x)′=−csc²x on their domains.

Write sec x=1/cos x and csc x=1/sin x. The quotient rule gives (sec x)′=sec x tan x and (csc x)′=−csc x cot x. The signs pair up: tangent and secant rules are positive forms; cotangent and cosecant carry minus signs. All input angles are in radians.

Do not evaluate these derivatives where the original trig function is undefined. For example tan and sec exclude cos x=0, while cot and csc exclude sin x=0. Reexpressing an identity is also a useful way to rebuild a forgotten formula.

Let's work through it

  1. Differentiate tan x=sin x/cos x with the quotient rule: [cos²x+sin²x]/cos²x. Reason: the derivative of cos contributes a second minus sign.
  2. Use sin²x+cos²x=1 to get 1/cos²x=sec²x. Reason: the Pythagorean identity simplifies the numerator.
  3. Retain cos x≠0. Reason: tangent and the quotient are undefined at those inputs.

Rebuild all four rules from sine and cosine

Assume x is measured in radians and use the known rules (sin x)′ = cos x, (cos x)′ = −sin x, and (1)′ = 0. Rewrite tan x = sin x/cos x, cot x = cos x/sin x, sec x = 1/cos x, and csc x = 1/sin x. For a quotient u/v, differentiate as (u′v − uv′)/v² only where the original denominator v is nonzero.

For tangent the numerator becomes cos²x + sin²x, so (tan x)′ = 1/cos²x = sec²x. For cotangent the numerator is −sin²x − cos²x, so (cot x)′ = −1/sin²x = −csc²x. For secant the constant numerator contributes zero: [0·cos x − 1·(−sin x)]/cos²x = sin x/cos²x = sec x tan x. For cosecant the same order gives [0·sin x − 1·cos x]/sin²x = −cos x/sin²x = −csc x cot x.

Retain the original domains: tan and sec require cos x ≠ 0, excluding x = π/2 + kπ; cot and csc require sin x ≠ 0, excluding x = kπ, for integer k. A simplified derivative does not add an input where the original function is absent. The minus signs come from the sine/cosine derivatives and the quotient numerator order, not from memorizing a list of signs.

FUN-3.B.3

Keep the numerator order and the original domain

Derive the tangent, cotangent, secant and cosecant rules from their sine/cosine identities. Angles are in radians.

On 0 < x < π radians, y = csc x decreases to (π/2, 1) then increases. It is unbounded near 0 and π. The horizontal tangent at (π/2, 1) is y = 1.
The completed sketch has a defined function value 1 and zero derivative at π/2. Neither endpoint is in the domain; the branches continue upward beyond the viewing window.
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On 0 < x < π radians, y = csc x decreases to (π/2, 1) then increases. It is unbounded near 0 and π. The horizontal tangent at (π/2, 1) is y = 1.

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  1. For tan x = sin x/cos x, the quotient rule gives [cos x·cos x − sin x·(−sin x)]/cos²x = (cos²x + sin²x)/cos²x = sec²x, where cos x ≠ 0.
  2. For cot x = cos x/sin x, it gives [(−sin x)·sin x − cos x·cos x]/sin²x = −(sin²x + cos²x)/sin²x = −csc²x, where sin x ≠ 0.
  3. For sec x = 1/cos x, it gives [0·cos x − 1·(−sin x)]/cos²x = sin x/cos²x = sec x tan x, where cos x ≠ 0.
  4. For csc x = 1/sin x, it gives [0·sin x − 1·cos x]/sin²x = −cos x/sin²x = −csc x cot x, where sin x ≠ 0.
  5. At x = π/2 the cosecant derivative is zero because its numerator cos(π/2) is zero while sin(π/2) = 1. At x = 0 its denominator is zero, and neither cosecant nor its derivative is defined there.

Answer and check

In radians: tan′ = sec² and sec′ = sec·tan on cos x ≠ 0; cot′ = −csc² and csc′ = −csc·cot on sin x ≠ 0. Zero derivative at a defined point is different from an undefined derivative at an excluded input.

FUN-3.B.3

Try it from memory

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Question 1 of 20 self-assessed

Question 1

Differentiate cot x.

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Retrieval 1

Differentiate cot x.

Worked answer and reason

−csc²x where sin x≠0.

Use the quotient rule on cos x/sin x.

Retrieval 2

Differentiate sec x+csc x.

Worked answer and reason

sec x tan x−csc x cot x on the common domain.

Use each reciprocal-trig rule with its correct sign.

Catch the tempting mistake

All four reciprocal-trig derivatives have positive signs.
Cotangent and cosecant derivatives have negative signs; derive them from identities if uncertain.
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Derive the complete reciprocal-trig suite

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Angles x are in radians. You may use (sin x)′ = cos x, (cos x)′ = −sin x, (1)′ = 0, the quotient rule, and sin²x + cos²x = 1. Derive the requested rules from identities rather than citing memorized tan, cot, sec or csc derivative formulas.

(a) Rewrite tan x, cot x, sec x and csc x as quotients involving only sin x, cos x and 1. Derive all four derivative rules by showing each quotient-rule numerator and denominator, simplify using identities, and state each original nonzero-denominator domain. State why the unscaled rules require radian inputs.

FUN-3.B.3

(b) Evaluate the four derivatives exactly at x = π/3. Then explain which original functions are undefined at x = 0 and at x = π/2, and why a simplified derivative formula cannot supply derivatives at excluded inputs.

FUN-3.B.3

(c) Use csc x = 1/sin x and its quotient-derived derivative to find the value and tangent slope at π/2. Sketch y = csc x on 0 < x < π with its tangent there, endpoint asymptotes, minimum point, and decreasing/increasing intervals. Explain why zero slope here is not an undefined derivative.

FUN-3.B.3

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Diagnose a quotient sign or domain error

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For radian x, csc x = 1/sin x and (sin x)′ = cos x. The quotient derivative is (u′v − uv′)/v².

(a) Choose the line that correctly derives the cosecant derivative from this identity while retaining the original function’s domain.

FUN-3.B.3

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For the teacher
  • Rebuild one rule from an identity before practicing the full set.
  • Use domain restrictions and graphs to check whether a derivative sign makes sense.

Unit planning and formative guidance →

Glossary and deeper reading · 7 refreshers

These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.

  • Trigonometric identity

    A trigonometric identity is an equality that holds for every input where both sides are defined. Identities let you rewrite expressions without changing their values on that shared domain. For example, sin²θ+cos²θ=1 follows from the unit circle. Any division used while rewriting an identity can introduce domain restrictions.

  • Derivative

    A derivative gives the instantaneous rate of change of a function with respect to its input. At a point, it is the finite limit of the difference quotient when that limit exists. For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6. A derivative function collects those rates across the inputs where the function is differentiable.

    Read more: 2.2 · Derivative →

  • Quotient rule

    The quotient rule differentiates a quotient where the denominator is nonzero and both functions are differentiable. It gives (f/g)′=(f′g−fg′)/g². For example, differentiating x/(x+1) produces 1/(x+1)² for x≠−1. The order of the subtraction and the denominator square both matter.

    Read more: 2.9 · Quotient rule →

  • Radian

    A radian measures an angle by dividing the intercepted arc length by the circle’s radius. One full turn is 2π radians, so π radians equals 180 degrees. For example, an arc with length equal to its radius subtends an angle of one radian. The familiar derivative formulas for sine and cosine use radian inputs.

  • Domain

    The domain of a function is the set of inputs for which the function is defined. An expression and its context can both restrict the allowed inputs. For example, the real function f(x)=1/(x−2) has domain all real numbers except 2. For a model of elapsed time, a sensible domain may also require t≥0.

  • Function

    A mathematical function assigns exactly one output to each input in its domain. Inputs can share an output, but one input cannot have two different outputs. For example, f(x)=x² assigns the output 9 to both 3 and −3. Function notation names the rule and its input; f(3) means the output when the input is 3.

  • Interval

    An interval is a set of real numbers containing every number between any two of its members. Parentheses exclude an endpoint, while square brackets include it. For example, (1,4] includes 4 and every number between 1 and 4, but excludes 1. An infinite endpoint always uses a parenthesis because infinity is not a real endpoint.

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