The following identities show the relationship between the trigonometric functions of a particular angle
`sin^2 theta + cos^2 theta = 1`
`tan^2 theta +1=sec^2 theta` and
`1+cot ^2 theta = csc ^2 theta`
The reason why they are called the Pythagorean Identities when the Pythagorean Theorem concerns the sides of a right triangle is because they are based on the theorem.
Mathematically, the theorem states `a^2 +b^2=c^2` where c is the hypotenuse in a right-triangle and / or `x^2+y^2=r^2` where r is the radius.
The radius (r)of a unit circle always equals 1 and is used to simplify trigonometric equations and expressions. As r=1, using the standard ratios for sin and cos,which are
`sin theta = (opp)/(hyp)` and `cos theta = (adj)/(hyp)`
we understand that `sin theta= y/1=y` and `cos theta = x/1=x`
if we perceive the Cartesian plane and the angle formed between the x-axis and the line in question (the radius), we can work from the Pythagorean Theorem using `x^2 +y^2 = r^2` .
In a triangle `a^2+b^2 =c^2` so similarly, we have a connection. To simplify the explanation, we will use the 3:4:5 triangle remembering that each one is a ratio, as with the unit circle.
`therefore (3/5)^2 + (4/5)^2 = c^2` where c is the hypotenuse. Simplified:
` 9/25 +16/25 = 1`
To equate this with the first identity:
`sin^2theta+cos^2theta=1` the connection is now apparent and the reasoning behind the "Pythagorean Identities" is explained.
Ans: Pythagorean Identities are based on the Pythagorean Theorem and the concept of the unit circle and the hypotenuse of one.