Pythagorean Identities
Introduction
As the name implies, Pythagorean identities come from the Pythagorean theorem. This theorem states that the square of the hypotenuse, or longest side, of any right-angled triangle, equals the sum of the squares of the other two sides, or legs. By using this theorem, Pythagorean identities can be obtained from trigonometric ratios.
What Are Pythagorean Identities?
Derived from the Pythagorean theorem, Pythagorean identities are significant identities in trigonometry. Numerous trigonometric problems where one trigonometric ratio is supplied and the other ratios need to be calculated are solved with the help of these identities. The most widely used Pythagorean identity is the fundamental one, which states the relationship between `\text{sin}` and `\text{cos}`.
- The relationship between `\text{sin}` and `\text{cos}` can be defined by `\text{sin}^2θ + \text{cos}^2θ = 1`
The following are the other two Pythagorean identities:
- The relationship between `\text{sec}` and `\text{tan}` can be defined by `1 + \text{tan}^2θ = \text{sec}^2θ`
- The relationship between `\text{cot}` and `\text{cosec}` can be defined by `1 + \text{cot}^2θ = \text{cosec}^2θ`
Pythagorean Trigonometric Identities
Algebraic operations allow us to write each one of them distinctly. Specifically, there are three ways to write these trig Pythagorean identities:
- `\text{sin}^2θ + \text{cos}^2θ = 1 ⇒ 1 - \text{sin}^2θ = \text{cos}^2θ ⇒ 1 - \text{cos}^2θ = \text{sin}^2θ`
- `1 + \text{tan}^2θ = \text{sec}^2θ ⇒ \text{sec}^2θ - 1 = \text{tan}^2θ ⇒ \text{sec}^2θ - \text{tan}^2θ = 1`
- `1 + \text{cot}^2θ = \text{cosec}^2θ ⇒ \text{cosec}^2θ - 1 = \text{cot}^2θ ⇒ \text{cosec}^2θ - \text{cot}^2θ = 1`
Deriving Pythagorean Identities
Proof of the Pythagorean Identity for `\text{Sine}` and `\text{Cosine}`
Consider a right triangle with angle `θ`. Let ` a ` be the length of the adjacent side, ` b ` be the length of the opposite side, and ` c ` be the length of the hypotenuse.
Using the definitions of `\text{sine}` and `\text{cosine}`:
`\sin(\theta) = \frac{b}{c} \quad \text{and} \quad \cos(\theta) = \frac{a}{c}`
Squaring both of these expressions:
`\sin^2(\theta) = \left(\frac{b}{c}\right)^2 \quad \text{and} \quad \cos^2(\theta) = \left(\frac{a}{c}\right)^2`
Adding these two equations:
`\sin^2(\theta) + \cos^2(\theta) = \left(\frac{b}{c}\right)^2 + \left(\frac{a}{c}\right)^2 = \frac{b^2 + a^2}{c^2}`
As ` a `, ` b `, and ` c ` are the sides of a right triangle, by the Pythagorean theorem, ` a^2 + b^2 = c^2 `. Substituting this into the equation above:
`\sin^2(\theta) + \cos^2(\theta) = \frac{c^2}{c^2} = 1`
Thus, the Pythagorean identity for `\text{sine}` and `\text{cosine}`, ` \sin^2(\theta) + \cos^2(\theta) = 1 `, is proved.
Proof of the Pythagorean Identity for `\text{Tangent}` and `\text{Secant}`
Consider the same right triangle with angle `θ`. Using the definitions of `\text{tangent}` and `\text{secant}`:
`\tan(\theta) = \frac{b}{a} \quad \text{and} \quad \sec(\theta) = \frac{c}{a}`
Squaring the `\text{tangent}`:
`\tan^2(\theta) = \left(\frac{b}{a}\right)^2`
Adding `1` to both sides of the equation:
`\tan^2(\theta) + 1 = \left(\frac{b}{a}\right)^2 + 1 = \frac{b^2 + a^2}{a^2}`
Since ` a^2 + b^2 = c^2 `:
`\tan^2(\theta) + 1 = \frac{c^2}{a^2}`
We know `\quad \sec(\theta) = \frac{c}{a}` making `\frac{c^2}{a^2} = \sec^2(\theta)`
Thus, the Pythagorean identity for `\text{sec}` and `\text{tan}`, ` \tan^2(\theta) + 1 = \sec^2(\theta) `, is proved.
Proof of the Pythagorean Identity for `\text{Cotangent}` and `\text{Cosecant}`:
To prove this identity, we'll start with the Pythagorean Identity for `\text{sine}` and `\text{cosine}` that we proved earlier.
\( \sin^2(\theta) + \cos^2(\theta) = 1 \)
Dividing both sides by ` \sin^2(\theta) `, we get
`\frac{\sin^2(\theta)}{\sin^2(\theta)} + \frac{\cos^2(\theta) }{\sin^2(\theta)}= \frac{1}{\sin^2(\theta)}`
\( 1 + \cot^2(\theta) = \csc^2(\theta) \)
Thus, the Pythagorean identity for `\text{cosec}` and `\text{cot}`, ` 1 + \cot^2(\theta) = \csc^2(\theta) `, is proved.
These proofs establish the Pythagorean identities, which are fundamental in trigonometry and have wide-ranging applications.
Applications of Pythagorean Identities
The Pythagorean identities have several important applications in trigonometry and mathematics in general.
`1`. Trigonometric Simplification
Pythagorean identities are frequently used to simplify trigonometric expressions. By replacing trigonometric functions with their equivalent expressions using the Pythagorean identities, complicated expressions can be converted into more manageable forms.
`2`. Trigonometric Equations
Pythagorean identities are useful in solving trigonometric equations. They allow us to rewrite trigonometric functions in terms of other trigonometric functions, which can sometimes lead to easier equations to solve or to identify solutions more readily.
`3`. Trigonometric Proofs
Pythagorean identities are essential tools in proving various trigonometric identities and formulas. They serve as building blocks for proving more complex trigonometric identities.
`4`. Trigonometric Integrals
Pythagorean identities can be utilized in trigonometric integration to simplify integrands or to rewrite integrals in terms of different trigonometric functions. This simplification often makes it easier to evaluate integrals or to recognize certain patterns that lead to solutions.
`5`. Geometric Applications
Pythagorean identities have geometric applications beyond trigonometry. They are used in geometry to establish relationships between lengths and angles in triangles, especially in right triangles. This includes applications in areas such as navigation, engineering, and physics.
`6`. Fourier Series and Transforms
Pythagorean identities play a pivotal role in many applications like signal processing, engineering, and physics. They help in manipulating trigonometric functions and simplifying expressions encountered in these mathematical analyses.
Overall, Pythagorean identities are versatile tools with numerous practical applications across various fields of mathematics and its applications. Their importance extends beyond trigonometry and geometry, making them essential concepts in advanced mathematical studies and real-world problem-solving.
Solved Examples
Example `1`: Simplify the expression ` \frac{1 + \tan^2(\theta)}{\csc^2(\theta)} `.
Solution:
Using the Pythagorean identity ` 1 + \tan^2(\theta) = \sec^2(\theta) `, we have:
`\frac{1 + \tan^2(\theta)}{\csc^2(\theta)} = \frac{\sec^2(\theta)}{\csc^2(\theta)}`
We know `\sec(\theta) = \frac{1}{\cos(\theta)}` and `\csc(\theta) = \frac{1}{\sin(\theta)} `
Hence we can write
`\frac{\sec^2(\theta)}{\csc^2(\theta)} = \frac{\sin^2(\theta)}{\cos^2(\theta)} = \tan^2(\theta)`
Hence ` \frac{1 + \tan^2(\theta)}{\csc^2(\theta)} ` simplifies to ` \tan^2(\theta) `.
Example `2`: Simplify the expression: ` (\cos x - \sin x)^2+(\cos x + \sin x)^2 `.
Solution:
To simplify the given expression `(\cos x - \sin x )^2 + (\cos x + \sin x)^2`, we'll first expand each squared term and then combine like terms:
`1`. Expand the squared terms:
\( (\cos x - \sin x)^2 = (\cos x - \sin x)(\cos x - \sin x) \)
\( = \cos^2x - 2\cos x\sin x + \sin^2x \)
\( (\cos x + \sin x)^2 = (\cos x + \sin x)(\cos x + \sin x) \)
\( = \cos^2x + 2\cos x\sin x + \sin^2x \)
`2`. Now, add the two expanded terms together:
\( (\cos^2x - 2\cos x\sin x + \sin^2x) + (\cos^2x + 2\cos x\sin x + \sin^2x) \)
\( = \cos^2x + \sin^2x - 2\cos x\sin x + \cos^2x + \sin^2x + 2\cos x\sin x \)
`3`. Combine like terms:
\( = 2\cos^2x + 2\sin^2x \)
Since `\cos^2x + \sin^2x = 1` (due to the Pythagorean identity for trigonometric functions), we can simplify further:
\( = 2(1) \)
\( = 2 \)
So, the simplified expression is `2`.
Example `3`: Prove the identity ` \sin^4(x) - \cos^4(x) = \sin^2(x) - \cos^2(x) `.
Solution:
Lets use the formula for difference of squares to write:
\( \sin^4(x) - \cos^4(x) = (\sin^2(x) + \cos^2(x))(\sin^2(x) - \cos^2(x)) \)
As per the Pythagorean identity for `\text{sine}` and `\text{cosine}`
\( \sin^2(x) + \cos^2(x) = 1 \)
Plugging in `1` for ` \sin^2(x) + \cos^2(x) = 1 ` we get
\( (\sin^2(x) + \cos^2(x))(\sin^2(x) - \cos^2(x)) = \sin^2(x) - \cos^2(x) \)
Hence proved that ` \sin^4(x) - \cos^4(x) = \sin^2(x) - \cos^2(x) `.
Example `4`: Simplify the expression ` \frac{\sin^2x + \tan^2x +\cos^2x}{\secx}`.
Solution:
As per Pythagorean identity for `\text{sin}` and `\text{cos}`, ` \sin^2x + \cos^2x = 1 `.
Using this we can write:
`\frac{\sin^2x + \tan^2x +\cos^2x}{\sec x} = \frac{1 + \tan^2x }{\sec x}`
As per Pythagorean identity for `\text{sec}` and `\text{tan}`, ` \sec^2x = 1+ \tan^2x`.
Using this we can write:
`\frac{1 + \tan^2x }{\sec x}\ = \frac{\sec^2x}{\sec x}`
Simplifying further
`\frac{\sec^2x}{\sec x} = \sec x`
Hence ` \frac{\sin^2x + \tan^2x +\cos^2x}{\sec x} ` simplifies to ` \sec x `.
Practice Problems
Q`1`. What is the Pythagorean identity for `\text{sine}` and `\text{cosine}`?
- ` \sin^2(\theta) + \cos^2(\theta) = 1 `
- ` \tan^2(\theta) + \cot^2(\theta) = 1 `
- ` \csc^2(\theta) + \sec^2(\theta) = 1 `
- ` \sin(\theta) \cdot \cos(\theta) = 1 `
Answer: a
Q`2`. What is the Pythagorean identity for `\text{tangent}` and `\text{secant}`?
- ` \tan^2(\theta) + 1 = \sec^2(\theta) `
- ` \sin^2(\theta) + \cos^2(\theta) = 1 `
- ` \cot^2(\theta) + \csc^2(\theta) = 1 `
- ` \tan(\theta) \cdot \sec(\theta) = 1 `
Answer: a
Q`3`. What will be the expression ` \frac{1- \sin^2x}{\cos x} ` in its simplest form?
- ` \cos x `
- ` \sin x `
- ` \tan x `
- ` \csc x `
Answer: a
Q`4`. Write the expression ` \frac{1- \sin^2(\theta)}{\cot^2(\theta)} ` in its simplest form.
- ` \cos^2(\theta) `
- ` \sin^2(\theta) `
- ` \frac{1}{\cos^2(\theta)} `
- ` \frac{1}{\sin^2(\theta)} `
Answer: b
Q`5`. What will be the expression ` \frac{\tan(\theta) + \cot(\theta)}{\tan(\theta)} ` in its simplest form?
- ` \csc^2(\theta) `
- ` \sin^2(\theta) `
- ` \sin(\theta)\cos(\theta) `
- ` \tan(\theta) `
Answer: a
Frequently Asked Questions
Q`1`. What are the Pythagorean identities in trigonometry?
Answer: The Pythagorean identities are fundamental trigonometric identities that relate the trigonometric functions of an angle in a right triangle. The three Pythagorean identities are:
- ` \sin^2(\theta) + \cos^2(\theta) = 1 `
- ` 1 + \tan^2(\theta) = \sec^2(\theta) `
- ` 1 + \cot^2(\theta) = \csc^2(\theta) `
Q`2`. How can I simplify trigonometric expressions using identities?
Answer: Trigonometric expressions can often be simplified by applying trigonometric identities. For example, you can use the Pythagorean identities to rewrite expressions in terms of `\text{sine}` and `\text{cosine}`, or you can use reciprocal and quotient identities to rewrite trigonometric functions in terms of other functions.
Q`3`. How do I prove trigonometric identities?
Answer: To prove trigonometric identities, you typically start with one side of the equation and manipulate it algebraically until you arrive at the other side of the equation. This may involve applying trigonometric identities, factoring, combining like terms, or using other properties of trigonometric functions.
Q`4`. What are some common strategies for proving trigonometric identities?
Answer: Some common strategies for proving trigonometric identities include:
- Starting with the more complex side of the equation and simplifying it step by step.
- Expressing trigonometric functions in terms of `\text{sine}` and `\text{cosine}` using Pythagorean identities.
- Using trigonometric identities such as double angle, half angle, sum and difference, and product-to-sum identities.
- Utilising algebraic techniques such as factoring, expanding, and combining like terms.
Q`5`. How can I use trigonometric identities to solve trigonometric equations?
Answer: Trigonometric identities can be used to simplify trigonometric equations and express them in terms of simpler functions. This often makes it easier to solve the equations by identifying patterns or using algebraic techniques. Additionally, trigonometric identities can help in verifying solutions obtained from solving trigonometric equations.