Why These Identities Matter
Cotangent and cosecant are linked by two useful facts: their derivatives have simple forms, and their squares satisfy a Pythagorean identity. Together, these facts help explain why certain factors in trigonometric integrals suggest a particular substitution.
For example, a factor of \(\csc^2 x\) is closely connected to the derivative of \(\cot x\), while a factor of \(\csc x\cot x\) is closely connected to the derivative of \(\csc x\). Recognizing these relationships is a matter of matching an integrand to the chain rule. The match must include the correct sign and any required constant factor.
Throughout, angles are measured in radians. The functions \(\cot x\) and \(\csc x\) are defined only where \(\sin x\ne 0\), so the derivative formulas below apply on intervals that do not contain an integer multiple of \(\pi\).
Definitions and Domains
The excluded points are \(x=k\pi\), where \(k\) is an integer. Between consecutive excluded points, both functions are defined and differentiable. When an antiderivative is stated on such an interval, its constant of integration may depend on the interval.
Deriving the Derivative Identities
To differentiate cotangent, use the quotient rule with numerator \(\cos x\) and denominator \(\sin x\). The derivative of the numerator is \(-\sin x\), and the derivative of the denominator is \(\cos x\). Therefore, wherever \(\sin x\ne 0\),
The penultimate step uses \(\sin^2 x+\cos^2 x=1\), and the final step uses the definition of cosecant. The negative sign is essential: cotangent decreases on each interval of its domain.
For cosecant, it is convenient to differentiate the reciprocal \(1/\sin x\). The reciprocal rule gives
This formula also has a negative sign. Both derivative formulas follow from standard differentiation rules and the definitions; neither is valid at points where \(\sin x=0\), because the functions themselves are undefined there.
The Pythagorean Identity
The identity connecting cotangent and cosecant follows directly from the Pythagorean identity for sine and cosine. Divide \(\sin^2 x+\cos^2 x=1\) by \(\sin^2 x\), which is allowed whenever \(\sin x\ne 0\). This gives
The identity is useful in both directions. A factor of \(\csc^2 x\) can be rewritten as \(1+\cot^2 x\), and a factor of \(1+\cot^2 x\) can be rewritten as \(\csc^2 x\). Which form is helpful depends on the rest of the integrand. The identity does not, by itself, determine a substitution; it gives an equivalent expression that may expose one.
Worked Example: Differentiate Cotangent from Its Definition
Worked Example: Differentiate Cotangent from Its Definition
Use the quotient rule to find the derivative of \(\cot x\), then compare the result with the derivative identity.
Write \(\cot x=\cos x/\sin x\). Applying the quotient rule gives
To check, replace \(\csc^2 x\) by \(1/\sin^2 x\). The result is \(-1/\sin^2 x\), exactly the quotient-rule derivative. This check also confirms that the derivative is negative.
How the Derivatives Guide Substitution
The substitution rule is the chain rule used in reverse. If an integrand contains a function of \(u(x)\) multiplied by \(u'(x)\), setting \(u=u(x)\) changes the integral into one in \(u\). In these trigonometric integrals, the derivative identities identify especially recognizable choices for \(u\).
Since \(d(\cot x)/dx=-\csc^2 x\), the substitution \(u=\cot x\) has differential \(du=-\csc^2 x\,dx\).
Since \(d(\csc x)/dx=-\csc x\cot x\), the substitution \(u=\csc x\) has differential \(du=-\csc x\cot x\,dx\).
The matching factor must be present, including its sign and any constant multiplier. If it is not present, the substitution does not immediately turn the entire integral into a function of \(u\) alone.
For power integrals, a power of the proposed substitution may remain after the derivative factor is separated. The identity \(\csc^2 x=1+\cot^2 x\) can then help express the remaining factors in terms of the chosen variable. These observations explain why the derivative formulas and identity are useful together. They are a guide to recognizing structure, not a license to change variables while leaving unmatched factors behind.
Worked Example: Integrate a Cosecant-Squared Factor
Worked Example: Integrate a Cosecant-Squared Factor
Find an antiderivative of \(\csc^2 x\) on an interval where \(\sin x\ne 0\).
The derivative identity says \(d(\cot x)/dx=-\csc^2 x\). Therefore the negative of cotangent has derivative \(\csc^2 x\):
Check by differentiating the proposed antiderivative:
The derivative equals the original integrand, so the result is correct on each interval in the domain. The constant \(C\) accounts for the family of antiderivatives on that interval.
Worked Example: Integrate a Cosecant-Cotangent Product
Worked Example: Integrate a Cosecant-Cotangent Product
Find an antiderivative of \(\csc x\cot x\) on an interval where \(\sin x\ne 0\).
The derivative of \(\csc x\) is \(-\csc x\cot x\). Thus the sign must be reversed to obtain the positive integrand:
Differentiate to check:
The derivative reproduces the integrand, confirming both the antiderivative and its sign.
Worked Example: Use the Identity to Integrate a Cotangent Square
Worked Example: Use the Identity to Integrate a Cotangent Square
Find an antiderivative of \(\cot^2 x\) on an interval where \(\sin x\ne 0\).
Rearrange \(\csc^2 x=1+\cot^2 x\) to obtain \(\cot^2 x=\csc^2 x-1\). Integrate each term:
Differentiate the proposed result:
The derivative matches the integrand. This short example illustrates how the identity can convert a power into a combination involving a derivative already known. It does not give a general reduction rule for arbitrary powers.
A Common Sign Error
A frequent mistake is to forget the negative sign when differentiating cotangent, then carry that error into an antiderivative. For example, the calculation
is wrong. Differentiating the proposed answer shows the problem:
which is the negative of the requested integrand. The correction is to use the negative cotangent:
The safest habit is to check an antiderivative by differentiation before accepting it. In these examples, that check catches the sign immediately.
Why These Relationships Are Useful
The derivative identities tell us which trigonometric factors can serve as the differential in a substitution. The Pythagorean identity lets us rewrite related powers so that the remaining factors can be considered in terms of the chosen variable. This is the basic structural reasoning behind many cotangent and cosecant integrals.
When deciding whether a substitution is appropriate, ask three questions: What expression could be called \(u\)? Is its derivative factor present? Can the other factors be rewritten in terms of \(u\)? A substitution is justified only when the transformed integrand is fully expressed in the new variable, with the differential accounted for. If any factor remains unexplained, the substitution has not yet finished the job.
Check Your Understanding
Use the derivative identities and the Pythagorean identity to answer these questions.
- Starting from \(\cot x=\cos x/\sin x\), which quotient-rule numerator produces the negative sign in its derivative?
- On what points are \(\cot x\) and \(\csc x\) undefined?
- What substitution is suggested by a factor of \(\csc^2 x\,dx\), and what is its differential?
- What is the derivative of \(\csc x\), including its sign and both trigonometric factors?
- Use \(\csc^2 x=1+\cot^2 x\) to rewrite \(\cot^2 x\) without cotangent.
