1. The
first Pythagorean Identity comes from the unit circle by drawing out a right
triangle and using all three sides of it: the hypotenuse, opposite side, and adjacent
side. The hypotenuse equals one (unit circle radius always equals 1), we
will label the opposite side with Y and the adjacent side with X. Now we plug
this into the Pythagorean Theorem Formula as Y^2+X^2=1. Now we
just substitute the Y with Sinᶿ and the X with Cos ᶿ, this leaves us to get our Pythagorean Identities Equation. The Pythagorean Identity equation is Sin^2ᶿ+Cos^2ᶿ=1.
This video I included shows the derivation of the First Pythagorean Identities followed by the second and third Pythagorean Identity. I included this video because it goes step by step using the Unit Circle Ratios and Reciprocal Identities to get the all Pythagorean Identities. Enjoy!!
The first identity is the base of the Pythagorean Identities because it
helps us find the other two Identities. To get to our second identity involving tanᶿ
and secᶿ we need to divide the first identity with cosine.
Cos^2ᶿ Cos^2ᶿ Cos^2ᶿ Cos^2ᶿ reciprocal identities
to obtain the
second identity
In the end
you get: tan^2ᶿ+1=sec^2ᶿ Don’t forget to
power up the identities.
In getting the
third identity we need to divide the first identity with sin.
Sin^2ᶿ Sin^2
Sin^2ᶿ Sin^2ᶿ reciprocal identities to obtain the 3rd identity
In
the end you get: 1+Cot^2ᶿ=Csc^2ᶿ
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