// maths › Trigonometric Identities
Verify and explore the three Pythagorean identities — sin²θ+cos²θ=1, 1+tan²θ=sec²θ and 1+cot²θ=cosec²θ — for any angle, in degrees or radians, with the full unit-circle derivation worked step by step.
sin²θ + cos²θ = 1
A mind behind this: Pythagoras of Samos c. 570 – c. 495 BC
They are three always-true relationships between the trig functions: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = cosec²θ. The first comes straight from Pythagoras' theorem applied to the unit circle, and the other two follow by dividing it through by cos²θ and sin²θ.
Take θ = 30°. Then sin 30° = 0.5 and cos 30° ≈ 0.8660. Squaring gives 0.25 and 0.75, and 0.25 + 0.75 = 1 exactly — the identity holds. Switch the toggle to radians and θ = 0.5236 rad gives the same result.
An equation is true only for particular values; an identity is true for every value of the variable. sin²θ + cos²θ = 1 holds for any angle θ at all, which is why it can be used to rewrite expressions freely.
Everywhere oscillations and rotations appear: AC electrical engineering (power factor), signal processing, computer graphics rotations, GPS and navigation, and physics — wherever sin and cos need to be swapped or simplified, these identities do the work.
The tan/sec form needs cos θ ≠ 0, and the cot/cosec form needs sin θ ≠ 0. At θ = 90°, for example, cos θ = 0, so sec and tan are undefined and the calculator flags this rather than returning a misleading number.