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.

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## 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:

- \(\color{blue}{sin^2\theta +cos^2\theta =1}\) (which gives the relation between \(sin\) and \(cos\))

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 \(sin\:\theta \:-\:cos\:\theta\)?

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

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