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Let us take a right angled triangle with hypotenuse length 1. If we mark one of the acute angles as , then using the definition of the sine ratio, we have

As the hypotenuse is 1,

Repeating the same process using the definition of the cosine ratio, we have

Pythagorean identities[edit | edit source]

Since this is a right triangle, we can use the Pythagorean Theorem:

This is the most fundamental identity in trigonometry.

From this identity, if we divide through by squared cosine, we are left with:

If instead we divide the original identity by squared sine, we are left with:

There are basically 3 main trigonometric identities. The proofs come directly from the definitions of these functions and the application of the Pythagorean theorem:

Angle sum-difference identities[edit | edit source]

Cofunction identities[edit | edit source]

Multiple angle identities[edit | edit source]