# Pythagorean Identities

The Pythagorean theorem can be applied to the trigonometric ratios that give rise to the Pythagorean identity. In this step-by-step guide, you will learn the concept of Pythagorean identity.

In mathematics, identity is an equation that holds for all possible values. An equation that contains trigonometric functions and is true for any value that replaces the variable is called trigonometric identity.

**Related Topics**

**A step-by-step** **guide to Pythagorean identities**

Pythagorean identities are important identities in trigonometry derived from the Pythagorean theorem. These identities are used to solve many trigonometric problems in which a trigonometric ratio is given and other ratios are found.

The fundamental Pythagorean identity shows the relationship between \(sin\) and \(cos\), and is the most common Pythagorean identity that says:

- \(\color{blue}{sin^2\theta +cos^2\theta =1}\)

There are two other Pythagorean identities as follows:

- \(\color{blue}{sec^2\theta -tan^2\theta =1}\) (which gives the relation between \(sec\) and \(tan\))
- \(\color{blue}{csc^2\theta -cot^2\theta =1}\) (which gives the relation between \(csc\) and \(cot\))

**Pythagorean trig identities**

All Pythagorean trig identities are listed below.

- \(\color{blue}{sin^2\theta +cos^2\theta =1}\)
- \(\color{blue}{1+tan^2\theta =sec^2\theta}\)
- \(\color{blue}{1+cot^2\theta =cosec^2\theta}\)

Each of them can be written in different forms with algebraic operations. That is, any Pythagorean identity can be written in three ways as follows:

- \(\color{blue}{sin^2θ + cos^2θ = 1 ⇒ 1 – sin^2θ = cos^2 θ ⇒ 1 – cos^2θ = sin^2θ}\)
- \(\color{blue}{sec^2θ\ – tan^2θ = 1 ⇒ sec^2θ = 1 + tan^2θ ⇒ sec^2θ – 1 = tan^2θ}\)
- \(\color{blue}{csc^2θ\ – cot^2θ = 1 ⇒ csc^2θ = 1 + cot^2θ ⇒ csc^2θ – 1 = cot^2θ}\)

**Pythagorean Identities** **– Example 1:**

In a right-angled triangle \(ABC\), angle \(C=90^{\circ }\), \(BAC = θ\), \(sin\:\theta = \frac{4}{5}\). Find the value of \(cos\:\theta\).

**Solution:**

Use the identity \(sin^2θ + cos^2θ =1\)

\((\frac{4}{5})^2+cos^2θ = 1\)

\(cos^2θ=1-(\frac{4}{5})^2\)

\(cos\:\theta ={\sqrt{1-\left(\frac{4}{5}\right)^2}}\)

\(=\sqrt{\frac{9}{25}}\)

\(=\frac{3}{5}\)

**Exercises for** **Pythagorean Identities**

- Suppose that \(sec\:\theta =\:-\frac{29}{20}\), what is the value of \(tan\:\theta\) if it is also negative?
- If \(sin\:\theta\) and \(cos\:\theta\) are the roots of the quadratic equation \(x^2+ px +1= 0\), find \(p\).
- If \(sin\:\theta \:cos\:\theta =\frac{1}{4}\), what is the value of \(sin\:\theta \:-\:cos\:\theta\)?

- \(\color{blue}{-\frac{21}{20}}\)
- \(\color{blue}{\pm \sqrt{3}}\)
- \(\color{blue}{\frac{\sqrt{2}}{2}}\)

## Related to This Article

### More math articles

- How Is the PERT Test Scored?
- How to Find Asymptotes: Vertical, Horizontal and Oblique
- Write a Ratio
- How to Prepare for the ISEE Upper-Level Math Test?
- How to Find the Area of Composite Shapes?
- The Ultimate 7th Grade MCA Math Course (+FREE Worksheets)
- 10 Must-Have Math Teacher Supplies
- Why Do I Need MATLAB Assignments Experts Help and Where One Get It
- The Ultimate 7th Grade SOL Math Course (+FREE Worksheets)
- 6th Grade NYSE Math FREE Sample Practice Questions

## What people say about "Pythagorean Identities - Effortless Math: We Help Students Learn to LOVE Mathematics"?

No one replied yet.