__5.1 The Elementary
Identities__

The elementary identities can be classified into several categories

**The Sine/Cosine identities**.

Each of the trigonometric functions can be expressed in terms of sine and cosine.

tan A = sin A / cos A

cot A = cos A / sin A

csc A = 1 / sin A

sec A = 1 / cos A

**The Pythagorean identities**

Recall that the intersection point of the terminal side of A and the unit circle has coordinates ( cos A, sin A ). Thus

cos^{2} A + sin^{2} A = 1 Dividing through by cos^{2} A and
applying the sine/cosine identities yields

1 + tan^{2} A
= sec^{2} A If the
first identity is divided by sin^{2} A, we get

cot^{2} A + 1 = csc^{2} A

**The Cofunction Identities**

Each trigonometric function has its corresponding
cofunction. Recall that the ‘co’ in
cofunction comes from ‘*complementary*’, as in ‘complementary angle’. The cofunction is always the function of the
complementary angle. The complement of
A is always ( π / 2 – A ). Thus

cos A = sin ( π / 2 – A )

sin A = cos ( π / 2 – A )

csc A = sec ( π / 2 – A )

sec A = csc ( π / 2 – A )

cot A = tan ( π / 2 – A )

tan A = cot ( π / 2 – A )