From Six Functions to Two
The six trigonometric functions are often introduced as separate ways to describe an angle. Their definitions, however, connect them closely: tangent and cotangent are quotients of sine and cosine, while secant and cosecant are reciprocals of cosine and sine.
These relationships are useful both for understanding the functions and for changing the form of an expression. The domain conditions matter: a quotient is defined only when its denominator is nonzero, and a reciprocal is defined only when the function being reciprocated is nonzero.
Definitions and Domain Conditions
Let \(x\) be an angle, measured consistently in degrees or radians. The sine and cosine functions are defined for every real angle. The other four functions are defined from sine and cosine, with division permitted only when the denominator is nonzero.
These conditions are part of the definitions, not optional qualifications. For example, tangent is undefined when cosine is zero; cotangent is undefined when sine is zero. Secant and cosecant have the corresponding restrictions.
| Function | Definition using sine and cosine | Condition |
|---|---|---|
| \(\tan x\) | \(\dfrac{\sin x}{\cos x}\) | \(\cos x\ne 0\) |
| \(\cot x\) | \(\dfrac{\cos x}{\sin x}\) | \(\sin x\ne 0\) |
| \(\sec x\) | \(\dfrac{1}{\cos x}\) | \(\cos x\ne 0\) |
| \(\csc x\) | \(\dfrac{1}{\sin x}\) | \(\sin x\ne 0\) |
Deriving the Quotient Identities
The quotient identities follow from the unit-circle definitions. At an angle \(x\), the unit-circle point has coordinates \((\cos x,\sin x)\). When \(\cos x\ne 0\), the ratio of the vertical coordinate to the horizontal coordinate is defined, and tangent is that ratio. Thus
This is the quotient identity for tangent. When \(\sin x\ne 0\), the reverse ratio, horizontal coordinate divided by vertical coordinate, is defined and gives cotangent:
Together these are called the quotient identities. Their denominators explain exactly where each identity applies. In particular, the cotangent quotient remains defined when \(\cos x=0\), provided \(\sin x\ne 0\); the tangent quotient does not.
Deriving the Reciprocal Identities
The definitions of secant and cosecant already show two reciprocal relations: secant is the reciprocal of cosine, and cosecant is the reciprocal of sine. Taking reciprocals of the quotient definitions also gives relations between tangent and cotangent, provided the functions involved are defined and nonzero.
For example, if \(\sin x\ne 0\) and \(\cos x\ne 0\), then
The conditions here ensure that \(\tan x\) exists and is nonzero, so its reciprocal exists. The same reasoning gives \(1/\cot x=\tan x\) when both are defined and nonzero.
The reciprocal identities for secant and cosecant follow directly from their definitions. For instance, when \(\cos x\ne 0\),
The second equality is valid because \(\sec x\ne 0\): its numerator is \(1\). Similarly, when \(\sin x\ne 0\), \(1/\csc x=\sin x\).
Why the Conditions Matter
An identity involving quotients is an equality only where both sides are defined. This can make two formulas that look reciprocal have different domains. Cotangent is defined whenever \(\sin x\ne 0\), including angles where \(\cos x=0\). But \(\tan x\) is undefined at those angles, so \(1/\tan x\) is undefined there too. Therefore, \(\cot x=1/\tan x\) is not a way to define cotangent at every angle in its domain.
The same principle applies to all four functions: check each denominator before dividing or taking a reciprocal. A useful way to justify a transformation is to name the condition that permits it.
Worked Examples
Worked Example: Compute All Four Functions from Sine and Cosine
Suppose an angle \(x\) has \(\sin x=\dfrac{3}{5}\) and \(\cos x=\dfrac{4}{5}\). Find \(\tan x\), \(\cot x\), \(\sec x\), and \(\csc x\).
Both given values are nonzero, so all four definitions apply. For tangent, use the sine-over-cosine quotient:
For cotangent, reverse the quotient:
For secant and cosecant, take the reciprocals:
Check the results against the defining relations. We have \(\tan x\cos x=(3/4)(4/5)=3/5=\sin x\), and \(\cot x\sin x=(4/3)(3/5)=4/5=\cos x\). Also, \(\sec x\cos x=(5/4)(4/5)=1\) and \(\csc x\sin x=(5/3)(3/5)=1\), as required.
Worked Example: Rewrite an Expression Using Sine and Cosine
Rewrite \(\sec x-\tan x\) using only sine and cosine, and evaluate the rewritten expression when \(\sin x=\dfrac{3}{5}\) and \(\cos x=\dfrac{4}{5}\).
Both terms require \(\cos x\ne 0\). Substitute their definitions, then combine the fractions, which have the same denominator:
Now substitute the given values into the original expression:
Check using the rewritten expression:
The two forms agree, and the condition \(\cos x\ne 0\) is retained because the original secant and tangent terms are undefined otherwise.
Worked Example: Rewrite a Product of Reciprocal Functions
Rewrite \((\cot x)(\csc x)\) using only sine and cosine. Then evaluate it when \(\sin x=\dfrac{3}{5}\) and \(\cos x=\dfrac{4}{5}\).
Both factors require \(\sin x\ne 0\). Substitute the quotient identity for cotangent and the reciprocal identity for cosecant:
For the given values, evaluate the original product:
Check by evaluating the rewritten form:
The values match, and the restriction \(\sin x\ne 0\) ensures that both original factors are defined.
Common Mistake: Reversing the Quotient
A frequent error is to write \(\tan x=\cos x/\sin x\), reversing the order in the quotient identity. The given values in the first example provide a direct check:
but the definition gives \(\tan x=3/4\). The value \(4/3\) is \(\cot x\), not \(\tan x\). Keep the order straight by recalling that tangent is sine divided by cosine, while cotangent is cosine divided by sine. As a final check, substitute the proposed tangent into \(\tan x\cos x=\sin x\): \((3/4)(4/5)=3/5\), which recovers the given sine.
Using the Identities Carefully
When rewriting an expression, proceed in a controlled order:
Decide which terms are tangent, cotangent, secant, or cosecant.
Replace each with its quotient or reciprocal in sine and cosine.
Check the denominators in the original expression before making a transformation.
Combine fractions or products only after checking that their denominators are nonzero.
These identities provide a common language for the six trigonometric functions: an expression involving all six can often be written using sine and cosine alone. In this tutorial, the goal is to establish that change of form while preserving the conditions under which the original expression exists.
Check Your Understanding
For each question, state any domain condition needed before using the identity.
- Write \(\tan x\) as a quotient of sine and cosine.
- Write \(\cot x\) using sine and cosine, and state its denominator condition.
- Rewrite \(\sec x+\csc x\) using only sine and cosine.
- When is \(\cot x=1/\tan x\) valid, and why is its domain condition stronger than the condition for defining cotangent?
- Rewrite \((\tan x)(\sec x)\) using only sine and cosine, and state where the expression is defined.
