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Pythagorean Identity

Sine squared plus cosine squared equals one.

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Pythagorean Identities

The Pythagorean Theorem works on right triangles.  If you consider the \begin{align*}x\end{align*}x coordinate of a point along the unit circle to be the cosine and the \begin{align*}y\end{align*}y coordinate of the point to be the sine and the distance to the origin to be 1 then the Pythagorean Theorem immediately yields the identity:

\begin{align*}& y^2+x^2=1\\ & \sin^2 x+\cos^2 x=1\end{align*}y2+x2=1sin2x+cos2x=1

An observant student may guess that other Pythagorean identities exist with the rest of the trigonometric functions.  Is \begin{align*}\tan^2 x+\cot^2 x=1\end{align*}tan2x+cot2x=1 a legitimate identity? 

Pythagorean Identities 

The proof of the Pythagorean identity for sine and cosine is essentially just drawing a right triangle in a unit circle, identifying the cosine as the \begin{align*}x\end{align*}x coordinate, the sine as the \begin{align*}y\end{align*}y coordinate and 1 as the hypotenuse.

\begin{align*}\cos^2 x+\sin^2 x=1\end{align*}cos2x+sin2x=1

Most people rewrite the order of the sine and cosine so that the sine comes first.

\begin{align*}\sin^2 x+\cos^2 x=1\end{align*}sin2x+cos2x=1

The two other Pythagorean identities are:

  • \begin{align*}1+\cot^2 x=\csc^2 x\end{align*}1+cot2x=csc2x
  • \begin{align*}\tan^2 x+1=\sec^2 x\end{align*}tan2x+1=sec2x

To derive these two Pythagorean identities, divide the original Pythagorean identity by \begin{align*}\sin^2 x\end{align*}sin2x and \begin{align*}\cos^2 x\end{align*}cos2x respectively.

To derive the Pythagorean identity \begin{align*}1+\cot^2 x=\csc^2 x\end{align*}1+cot2x=csc2x divide through by \begin{align*}\sin^2 x\end{align*}sin2x and simplify.

\begin{align*}\frac{\sin^2 x}{\sin^2 x}+\frac{\cos^2 x}{\sin^2 x} &= \frac{1}{\sin^2 x}\\ 1+\cot^2 x &= \csc^2 x\end{align*}sin2xsin2x+cos2xsin2x1+cot2x=1sin2x=csc2x

Similarly, to derive the Pythagorean identity \begin{align*}\tan^2 x+1=\sec^2 x\end{align*}tan2x+1=sec2x, divide through by \begin{align*}\cos^2 x\end{align*}cos2x and simplify.

\begin{align*}\frac{\sin^2 x}{\cos^2 x}+\frac{\cos^2 x}{\cos^2 x}&= \frac{1}{\cos^2 x}\\ \tan^2 x+1 &= \sec^2 x\end{align*}sin2xcos2x+cos2xcos2xtan2x+1=1cos2x=sec2x

Examples

Example 1

Earlier, you were asked if \begin{align*}\tan^2 x+\cot^2 x=1\end{align*}tan2x+cot2x=1 is a legitimate identity. Cofunctions are not always connected directly through a Pythagorean identity.

\begin{align*}\tan^2 x+\cot^2 x \neq 1\end{align*}tan2x+cot2x1

Visually, the right triangle connecting tangent and secant can also be observed in the unit circle.  Most people do not know that tangent is named “tangent” because it refers to the distance of the line tangent from the point on the unit circle to the \begin{align*}x\end{align*}x axis.  Look at the picture below and think about why it makes sense that \begin{align*}\tan x\end{align*}tanx and \begin{align*}\sec x\end{align*}secx are as marked. \begin{align*}\tan x=\frac{opp}{adj}\end{align*}tanx=oppadj.  Since the adjacent side is equal to 1 (the radius of the circle), \begin{align*}\tan x\end{align*}tanx simply equals the opposite side.  Similar logic can explain the placement of \begin{align*}\sec x\end{align*}secx.

Example 2

Simplify the following expression: \begin{align*}\frac{\sin x (\csc x-\sin x)}{1-\sin x}\end{align*}sinx(cscxsinx)1sinx.

\begin{align*}\frac{\sin x (\csc x-\sin x)}{1-\sin x} &= \frac{\sin x \cdot \csc x-\sin^2 x}{1-\sin x}\\ &= \frac{1-\sin^2 x}{1-\sin x}\\ &= \frac{(1-\sin x)(1+\sin x)}{1-\sin x}\\ &= 1+\sin x\end{align*}sinx(cscxsinx)1sinx=sinxcscxsin2x1sinx=1sin2x1sinx=(1sinx)(1+sinx)1sinx=1+sinx

Note that factoring the Pythagorean identity is one of the most powerful applications.  This is very common and is a technique that you should feel comfortable using. 

Example 3

Prove the following trigonometric identity: \begin{align*}(\sec^2 x+\csc^2 x)-(\tan^2 x+\cot^2 x)=2\end{align*}(sec2x+csc2x)(tan2x+cot2x)=2

Group the terms and apply a different form of the second two Pythagorean identities which are \begin{align*}1+\cot^2 x=\csc^2 x\end{align*}1+cot2x=csc2x and \begin{align*}\tan^2 x+1=\sec^2 x\end{align*}tan2x+1=sec2x.

\begin{align*}(\sec^2 x+\csc^2 x)-(\tan^2 x+\cot^2 x) &= \sec^2 x-\tan^2 x+\csc^2 x-\cot^2 x\\ &= 1+1\\ &= 2\end{align*}(sec2x+csc2x)(tan2x+cot2x)=sec2xtan2x+csc2xcot2x=1+1=2

Example 4

Simplify the following expression.

\begin{align*}(\sec^2 x)(1-\sin^2 x)-\left(\frac{\sin x}{\csc x}+\frac{\cos x}{\sec x}\right)\end{align*}

\begin{align*}&(\sec^2 x)(1-\sin^2 x)-\left(\frac{\sin x}{\csc x}+\frac{\cos x}{\sec x}\right)\\ &=\sec^2 x \cdot \cos^2 x-(\sin^2 x+\cos^2 x)\\ &= 1-1\\ &= 0\end{align*}

Example 5

Simplify the following expression.

\begin{align*}(\cos t-\sin t)^2+(\cos t+\sin t)^2\end{align*}

Note that initially, the expression is not the same as the Pythagorean identity.

\begin{align*}&(\cos t-\sin t)^2+(\cos t+\sin t)^2 \\ &= \cos^2 t-2 \cos t \sin t+\sin^2 t +\cos^2 t+2 \cos t \sin t+\sin^2 t\\ &= 1-2 \cos t \sin t+1+2 \cos t \sin t\\ &= 2\end{align*}

Review

Prove each of the following:

1. \begin{align*}(1-\cos^2 x)(1+\cot^2 x)=1\end{align*}

2. \begin{align*}\cos x (1-\sin^2 x)=\cos^3 x\end{align*}

3. \begin{align*}\sin^2 x=(1-\cos x)(1+\cos x)\end{align*}

4. \begin{align*}\sin x=\frac{\sin^2 x+\cos^2 x}{\csc x}\end{align*}

5. \begin{align*}\sin^4 x-\cos^4 x=\sin^2 x-\cos^2 x\end{align*}

6. \begin{align*}\sin^2 x \cos^3 x=(\sin^2 x-\sin^4 x)(\cos x)\end{align*}

Simplify each expression as much as possible.

7. \begin{align*}\tan^3 x \csc^3 x\end{align*}

8. \begin{align*}\frac{\csc^2 x-1}{\sec^2 x}\end{align*}

9. \begin{align*}\frac{1-\sin^2 x}{1+\sin x}\end{align*}

10. \begin{align*}\sqrt{1-\cos^2 x}\end{align*}

11. \begin{align*}\frac{\sin^2 x-\sin^4 x}{\cos^2 x}\end{align*}

12. \begin{align*}(1+\tan^2 x)(\sec^2 x)\end{align*}

13. \begin{align*}\frac{\sin^2 x+\tan^2 x+\cos^2 x}{\sec x}\end{align*}

14. \begin{align*}\frac{1+\tan^2 x}{\csc^2 x}\end{align*}

15. \begin{align*}\frac{1-\sin^2 x}{\cos x}\end{align*}

Review (Answers)

To see the Review answers, open this PDF file and look for section 6.2. 

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Vocabulary

TermDefinition
identity An identity is a mathematical sentence involving the symbol “=” that is always true for variables within the domains of the expressions on either side.
Pythagorean Identity The Pythagorean identity is a relationship showing that the sine of an angle squared plus the cosine of an angle squared is equal to one.
Pythagorean Theorem The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by a^2 + b^2 = c^2, where a and b are legs of the triangle and c is the hypotenuse of the triangle.

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