What are Pythagorean Identities

Introduction:
                   In a unit circle, the study shows that a point on the unit circle (the vertex of the right triangle) can be represented by the coordinates.  (cos Θ, sin Θ).
                   Since the bases of a right triangle in the unit circle have the values of sinΘ and cosΘ, the Pythagorean Theorem can be used to obtain,
                                            sin²Θ + cos²Θ  = 1
This equation is called as Pythagorean identity. Here the value of theta in immaterial.

Second Pythagorean Identity:

                   Using the Pythagorean identity sin²Θ + cos²Θ = 1, we can derive two additional Pythagorean identities. Using the first Pythagorean identity
                                      sin²Θ + cos²Θ = 1
                   Divide each term by cos²Θ,
                                      sin²Θ/ cos²Θ + cos²Θ/cos²Θ = 1/cos²Θ
                   We know, sin²Θ/ cos²Θ = tan²Θ and 1/cos²Θ = sec²Θ
                                      So the second Pythagorean identity is, tan²Θ + 1 = sec²Θ

Third Pythagorean Identity:

                   We know, sin²Θ + cos²Θ = 1
                   Divide each term by, sin²Θ
                   We get,
                                      sin²Θ/sin²Θ + cos²Θ/sin²Θ = 1/sin²Θ
                   We know that, cos²Θ/sin²Θ = cot²Θ and 1/sin²Θ = csc²Θ
                                      So the third Pythagorean Identity is, 1 + cot²Θ = csc²Θ


Hope you liked the above explanation. Please leave your comments, if you have any doubts.