The identity `sin^2A + cos^2A = 1` has to be proved.
For any angle A it is possible to construct a right triangle ABC such that A is the angle under consideration, AB and BC are the smaller sides and AC is the hypotenuse.
sin A = (opposite side)/(hypotenuse) = `(BC)/(AC)`
cos A = (adjacent side)/(hypotenuse) = `(AB)/(AC)`
`sin^2 A + cos^2A`
= `(BC)^2/(AC)^2 + (AB)^2/(AC)^2`
=> `((BC)^2 + (AB)^2)/(AC)^2`
Applying Pythagoras' Theorem in the right triangle ABC gives `(AB)^2 + (BC)^2 = (AC)^2`
This proves that `sin^2 A + cos^2A = 1`
It can be done through unit circle method. But using pythogorous therorem would be easy.
Let us consider a right angled triangle with one of the angle as "A" which is formed by "l" as opposite and "b" as adjacent.
Sin A = opposite / hypotenuse
Cos A = adjacent / hypotenuse
(sinA)^2 + (cos A)^2 = (opposite / hypotenuse )^2 + (adjacent / hypotenuse)^2
=hypotenuse^2/hypotenuse^2 [as opposite^2+adjacent^2 = hypotenuse^2 ---pythogorous therorem]
(sin A)^2 + (Cos A)^2 = 1