Pythagorean trigonometric identity

The identity sin²θ + cos²θ = 1, expressing the unit-circle relation between sine and cosine. Dividing it by sine²θ or cosine²θ yields identities involving tangent, cotangent, secant, and cosecant.

Unit circle: The circle of radius one centered at the origin of a coordinate plane. A point on it has coordinates cos θ and sin θ, whose squared lengths sum to one.
Trigonometric functions: Functions that relate angles to ratios or coordinates, including sine, cosine, and tangent. The identity is a relation among particular trigonometric functions.
Trigonometric simplification: The rewriting of trigonometric expressions into equivalent forms suited to calculation or proof. Replacing sin²θ + cos²θ with one often shortens expressions.
Law of cosines: A triangle relation connecting side lengths and the cosine of an included angle. Unlike the Pythagorean identity, it relates three side lengths in a general triangle.
Sine and cosine: The trigonometric functions that give the vertical and horizontal coordinates of a unit-circle point. Their coordinate roles make the identity a direct consequence of the unit circle's equation.
Right triangle: A triangle containing one angle of 90 degrees. Its side ratios define sine and cosine, while the Pythagorean theorem links those ratios.
Trigonometric equations: Equations containing trigonometric functions whose angle solutions must be determined. The identity can reduce powers or convert functions to solve for possible angles.
Euler's formula: The equation e^(ix) = cos x + i sin x connecting complex exponentials with trigonometric functions. Its unit-modulus consequence gives another route to the sine-cosine identity.
Tangent function: The trigonometric function defined by tan θ = sin θ / cos θ where cosine is nonzero. Dividing the identity by cos²θ gives 1 + tan²θ = sec²θ.
Pythagorean theorem: The theorem stating that the squares of a right triangle's legs sum to the square of its hypotenuse. Dividing this theorem by the hypotenuse squared yields the sine-cosine identity.

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TangentRelated: Dividing it by cos²(x) yields 1 + tan²(x) = sec²(x).