The given identity, cos(x)² + sin(x)² = 1, is derived from the identity, cos(x+y) = cos(x)•cos(y) - sin(x)•sin(y), when y = -x, such that cos(x+y) = cos(0) = 1 = cos(x)² + sin(x)²
How can the relationships between identities be found?
From the cosine of the sum of two angles, we have;
cos(x+y) = cos(x)•cos(y) - sin(x)•sin(y)
Therefore;
cos(x+x) = cos(x)•cos(x) - sin(x)•sin(x)
Which gives;
cos(x+(-x)) = cos(x)•cos(-x) - sin(x)•sin(-x)
However;
cos(-x) = cos(x)
sin(-x) = -sin(x)
Therefore;
cos(x+(-x)) = cos(x)•cos(x) - sin(x)•(-sin(x))
cos(x+(-x)) = cos(x)² + sin(x)²
However;
cos(x+(-x)) = cos(x-x) = cos(0) = 1
Therefore;
cos(x+(-x)) = 1 = cos(x)² + sin(x)²
From which we have;
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