Even, Odd, and Cofunction Identities

Turning by −θ instead of θ reflects the terminal point across the x-axis. The x-coordinate survives; the y-coordinate flips.

The consequence

Cosine is even, so cos(−θ) = cos θ. Sine is odd, so sin(−θ) = −sin θ. Tangent is odd too.

Cofunctions

In a right triangle the two acute angles total 90°, and one angle’s opposite side is the other’s adjacent side. So sin θ = cos(90° − θ).

Where the co comes from

Cosine means the sine of the complement. The prefix is short for complementary, which is exactly what the identity says.

What they are for

They clear negative angles out of an expression and swap one function for another when only one is convenient.

Symmetry gives identities for free

Cosine is even: cos(−θ) = cos θ, because reflecting across the x-axis leaves the x-coordinate alone. Sine is odd: sin(−θ) = −sin θ, because the y-coordinate flips. Both follow from the picture, not from algebra.

Tangent is odd

Since tangent is sine over cosine, an odd function over an even one, it is odd: tan(−θ) = −tan θ. The parity of a quotient follows from the parities of its parts, which saves checking each case separately.

Cofunction identities

sin θ = cos(90° − θ) and similarly for the other pairs. In a right triangle the two acute angles are complementary, and one angle's opposite is the other's adjacent — which is exactly what the identity records.

That is where the co- comes from

Cosine is the sine of the complement; cotangent the tangent of the complement. The prefix is short for complement, which makes the naming of the whole family suddenly systematic rather than arbitrary.