Pythagorean Trigonometric Identity

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A Pythagorean Trigonometric Identity is the Pythagorean theorem expressed in terms of trigonometric functions.

[math]\displaystyle{ a^2+b^2=c^2 \Longleftrightarrow \left(\frac{a}{c}\right)^2+\left(\frac{b}{c}\right)^2=1 \Longleftrightarrow \sin^2(\theta)+\cos^2(\theta)=1 }[/math]

where c is the hypotenuse of the right triangle, a and b are the opposite and the adjacent side to the acute angle [math]\displaystyle{ \theta }[/math]
[math]\displaystyle{ 1 + \tan^2(\theta)= \sec^2(\theta)\quad\text{and}\quad 1 + \cot^2(\theta) = = \csc^2(\theta) }[/math]
where [math]\displaystyle{ \tan(\theta) }[/math] is the tangent function, [math]\displaystyle{ sec(\theta) }[/math] the secant function [math]\displaystyle{ \cot(\theta) }[/math] is the cotangent function and [math]\displaystyle{ \csc(\theta) }[/math] is the cosecant function.


References

2015

The identity is given by the formula:
[math]\displaystyle{ \sin^2 \theta + \cos^2 \theta = 1.\! }[/math]
(Note that sin2 θ means (sin θ)2). This relation between sine and cosine is sometimes called the fundamental Pythagorean trigonometric identity
If the length of the hypotenuse of a right triangle is 1, then the length of either of the legs is the sine of the opposite angle and is also the cosine of the adjacent acute angle. Therefore, this trigonometric identity follows from the Pythagorean theorem.