College Algebra & Trigonometry, 2018a

10.1. RECIPROCAL AND PYTHAGOREAN IDENTITIES 439
Substituting this equality gives us the first Pythagorean Identity:
x 2 + y 2 =1
or
cos 2 θ +sin 2 θ =1
This identity is usually stated in the form:
sin 2 θ +cos 2 θ =1
If we take this identity and divide through on both sides by cos 2 θ, this will result
in the first of two additional Pythagorean Identities:
sin 2 θ
cos 2 θ + cos2 θ
cos 2 θ = 1
cos 2 θ
or
tan 2 θ + 1 = sec 2 θ
Dividing through by sin 2 θ gives us the second:
sin 2 θ
sin 2 θ + cos2 θ
sin 2 θ = 1
sin 2 θ
or
1+cot 2 θ = csc 2 θ
So, the three Pythagorean Identities we will be using are:
sin 2 θ +cos 2 θ =1
tan 2 θ + 1 = sec 2 θ
1+cot 2 θ = csc 2 θ