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Imagine typing a monstrous expression like cos⁑θtan⁑θsin⁑θ\frac{\cos \theta \tan \theta}{\sin \theta} into a calculator, only to find it gives the exact value of 11 for every single angle on Earth.
This is not a coincidence β€” it is a trigonometric identity, an equation that remains true for every allowable input in its domain.

The Fundamental Toolkit

On a unit circle of radius 11, every point is given by (x,y)=(cos⁑θ,sin⁑θ)(x, y) = (\cos \theta, \sin \theta). From these basic coordinates, all other trigonometric functions emerge as simple ratios.
The reciprocal identities define csc⁑θ=1sin⁑θ\csc \theta = \frac{1}{\sin \theta}, sec⁑θ=1cos⁑θ\sec \theta = \frac{1}{\cos \theta}, and cot⁑θ=1tan⁑θ\cot \theta = \frac{1}{\tan \theta}, while the quotient identities give tan⁑θ=sin⁑θcos⁑θ\tan \theta = \frac{\sin \theta}{\cos \theta} and cot⁑θ=cos⁑θsin⁑θ\cot \theta = \frac{\cos \theta}{\sin \theta}.
What happens when we apply the Pythagorean theorem to that very same unit circle triangle?

The Pythagorean Identities

Substituting the unit circle coordinates (x,y)=(cos⁑θ,sin⁑θ)(x, y) = (\cos \theta, \sin \theta) into the unit circle equation x2+y2=1x^2 + y^2 = 1 establishes The Fundamental Pythagorean Identity: sin⁑2θ+cos⁑2θ=1\sin^2 \theta + \cos^2 \theta = 1.
Dividing by cos⁑2θ\cos^2 \theta yields 1+tan⁑2θ=sec⁑2θ1 + \tan^2 \theta = \sec^2 \theta, and dividing by sin⁑2θ\sin^2 \theta yields 1+cot⁑2θ=csc⁑2θ1 + \cot^2 \theta = \csc^2 \theta.
Rearranged forms (such as 1βˆ’sin⁑2ΞΈ=cos⁑2ΞΈ1 - \sin^2\theta = \cos^2\theta and sec⁑2ΞΈβˆ’1=tan⁑2ΞΈ\sec^2\theta - 1 = \tan^2\theta) are frequently applied in substitutions to simplify complex expressions.
Rearranged forms (e.g., 1βˆ’sin⁑2ΞΈ=cos⁑2ΞΈ1 - \sin^2 \theta = \cos^2 \theta and sec⁑2ΞΈβˆ’1=tan⁑2ΞΈ\sec^2 \theta - 1 = \tan^2 \theta) are frequently applied in substitutions to collapse complicated binomials.