Deriving the Pythagorean Identity

Why These Identities Hold

The sine and cosine of an angle are coordinates of a point on the unit circle. Every point on that circle satisfies the same geometric equation, so its coordinates give a relationship between sine and cosine that is true for every angle.

That relationship is the starting point for three important trigonometric identities. The first applies to all real angles. The other two follow by division, so each has a restriction: the quantity being divided by must be nonzero.

The Unit-Circle Equation

The unit circle is the circle of radius \(1\) centered at the origin. A point \((u,v)\) on it has distance \(1\) from the origin. By the distance formula, its coordinates satisfy \(u^2+v^2=1\). For an angle \(x\), the corresponding unit-circle point has coordinates \((\cos x,\sin x)\).

Substitute \(u=\cos x\) and \(v=\sin x\) into the circle equation. This gives the Pythagorean identity for sine and cosine.

The conventional order writes the sine term first. The two terms can be reversed because addition is commutative.

The identity holds for every real angle because every angle selects a point on the unit circle, and every point on that circle satisfies its defining equation. The signs of sine and cosine may change in different quadrants, but squaring removes those signs; the sum of the squares remains \(1\).

Deriving the Tangent-Secant Identity

The earlier definitions of tangent and secant are \(\tan x=\frac{\sin x}{\cos x}\) and \(\sec x=\frac{1}{\cos x}\). Both expressions require \(\cos x\ne 0\). When that condition holds, divide every term of the sine-cosine identity by \(\cos^2 x\).

The first term is \(\tan^2 x\), the second is \(1\), and the right-hand side is \(\sec^2 x\). Therefore:

The condition matters: tangent and secant are undefined wherever \(\cos x=0\). The identity is not a claim that either side has a value at those angles.

Deriving the Cotangent-Cosecant Identity

The definitions of cotangent and cosecant are \(\cot x=\frac{\cos x}{\sin x}\) and \(\csc x=\frac{1}{\sin x}\). These expressions require \(\sin x\ne 0\). Divide the sine-cosine identity by \(\sin^2 x\) when that condition holds:

The terms become \(1\), \(\cot^2 x\), and \(\csc^2 x\), respectively. Hence:

This identity is unavailable at angles where \(\sin x=0\), because cotangent and cosecant are undefined there.

Why the Derivations Work

Each derived identity comes from the same equation and a permitted division. The nonzero conditions are not optional technical details: division by zero is undefined. The steps below make the logic explicit.

This also explains the different domains. The sine-cosine identity itself requires no division, so it holds at every real angle. Each of the other identities uses a quotient with a denominator that must be nonzero.

Worked Examples

Common Pitfall: Dividing Without Checking

A frequent error is to divide the sine-cosine identity by \(\cos^2 x\) and then claim that \(\tan^2 x+1=\sec^2 x\) holds for every real \(x\). This overlooks the condition \(\cos x\ne 0\).

For example, at \(x=\frac{\pi}{2}\), \(\cos x=0\). The attempted division would include \(\frac{0}{0}\), which is undefined. In fact, \(\tan\frac{\pi}{2}\) and \(\sec\frac{\pi}{2}\) are undefined, so the tangent-secant identity cannot be evaluated there. The correction is to state its domain condition each time: it applies only when \(\cos x\ne 0\). The same care is needed when dividing by \(\sin^2 x\).

Using the Identities Carefully

The identities are useful for checking values and relating functions, but the squared equations do not determine signs on their own. For instance, \(\cos^2 x=\frac{16}{25}\) implies \(\cos x=\frac{4}{5}\) or \(\cos x=-\frac{4}{5}\). The quadrant or other information is needed to select the correct value.

Keep the three statements and their conditions distinct:

The first identity is the geometric foundation. The other two are consequences obtained by division, and their restrictions follow directly from the definitions of the functions involved.