4. Pythagorean identity

In trigonometry, the basic relationship between the sine and the cosine is known as the Pythagorean identity:

${\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1\!}$

where cos² θ means (cos(θ))² and sin² θ means (sin(θ))².

This can be viewed as a version of the Pythagorean theorem, and follows from the equation x² + y² = 1 for the unit circle. This equation can be solved for either the sine or the cosine:

${\displaystyle {\begin{aligned}\sin \theta &=\pm {\sqrt {1-\cos ^{2}\theta }},\\\cos \theta &=\pm {\sqrt {1-\sin ^{2}\theta }}.\end{aligned}}}$

where the sign depends on the quadrant of θ.

Related identities

Dividing the Pythagorean identity by either cos² θ or sin² θ yields two other identities:

${\displaystyle 1+\tan ^{2}\theta =\sec ^{2}\theta \quad {\text{and}}\quad 1+\cot ^{2}\theta =\csc ^{2}\theta .\!}$

Using these identities together with the ratio identities, it is possible to express any trigonometric function in terms of any other (up to a plus or minus sign):

Each trigonometric function in terms of the other five