The discussion centers on the identity ##\sin^2 x + \cos^2 x = 1## and its applicability to multiples of x, specifically examining whether this identity holds for expressions like ##\sin^2 2x + \cos^2 2x##, ##\sin^2 3x + \cos^2 3x##, and ##\sin^2 4x + \cos^2 4x##. The conversation includes both verification of the identity and exploration of related trigonometric identities.
Participants generally agree on the validity of the identity for multiples of x, but there are varying degrees of exploration regarding the implications and related identities, particularly concerning the double angle formulas.
Some participants reference specific trigonometric identities without fully resolving the implications or providing detailed proofs, leaving certain assumptions and steps in the discussion open-ended.
##sin^2(\text{whatever}) + cos^2(\text{whatever}) = 1##basty said:I know that ##\sin^2 x + cos^2 x = 1.##
Is this mean that
##\sin^2 2x + \cos^2 2x = 1##
or
##\sin^2 3x + \cos^2 3x = 1##
or
##\sin^2 4x + \cos^2 4x = 1##
and so on?
##\sin^2(A) = \frac{1 - \cos(2A)}{2}##basty said:Does ##\sin^2 2x = \frac{1 - \cos 4x}{2}?##