The Pythagorean Trigonometric Identity (College Board AP® Precalculus): Revision Note

The Pythagorean identity

What is the Pythagorean trigonometric identity?

• The Pythagorean trigonometric identity is:

${\mathbf{sin}}^{\mathbf{2}}\mathit{\theta }\mathbf{+}{\mathbf{cos}}^{\mathbf{2}}\mathit{\theta }\mathbf{=}\mathbf{1}$

• This holds for every value of $\theta$

  • It is a true statement for all real numbers

• The identity comes directly from the Pythagorean Theorem applied to the unit circle

  • For any angle $\theta$ in standard position, the terminal ray meets the unit circle at the point $(\cos \theta ,\sin \theta )$

  • The right triangle formed has legs of length $\left|\cos \theta \right|$ and $\left|\sin \theta \right|$

    • and hypotenuse $1$ (because the hypotenuse is the radius of the circle)

  • Applying the Pythagorean Theorem gives ${\cos }^2\theta +{\sin }^2\theta =1$

What useful rearrangements of the Pythagorean identity exist?

• The identity can be rearranged in several useful ways

  • These rearrangements are essential for rewriting trigonometric expressions

• Simply rearranging the terms in the identity

  • gives the two forms

    • ${\sin }^2\theta =1-{\cos }^2\theta$

    • ${\cos }^2\theta =1-{\sin }^2\theta$

• Dividing the original identity by ${\mathbf{cos}}^{\mathbf{2}}\mathit{\theta }$

  • gives a form involving tangent and secant

    • ${\tan }^2\theta +1={\sec }^2\theta$

  • which can also be written as

    • ${\sec }^2\theta -1={\tan }^2\theta$

• Dividing the original identity by ${\mathbf{sin}}^{\mathbf{2}}\mathit{\theta }$

  • gives a form involving cotangent and cosecant:

    • $1+{\cot }^2\theta ={\csc }^2\theta$

  • which can also be written as

    • ${\csc }^2\theta -1={\cot }^2\theta$

How is the Pythagorean identity used to rewrite trigonometric expressions?

• A common task is to rewrite a trigonometric expression as a single term involving only one specified function

• When approaching a question like this

  • Look for combinations like $1-{\sin }^{2}x$, ${\sec }^{2}x-1$, etc.

    • These can be replaced using one of the Pythagorean identities

  • If needed, convert any remaining reciprocal trig functions ($\sec$, $\csc$, $\cot$) into expressions involving $\sin$ and $\cos$

    • Similarly you can convert regular trig functions into reciprocal ones if necessary

  • Simplify the resulting expression algebraically

    • Cancelling factors, combining fractions, etc.

• These sorts of steps will let you convert the final expression back into the requested form

  • e.g. an expression involving $\tan x$ only