Explanation:
In the right triangle shown in the attachment, we have, by definition, ...
sin(θ) = a/c
cos(θ) = b/c
Then ...
sin²(θ) +cos²(θ) = (a/c)² +(b/c)² = (a² +b²)/c²
By the Pythagorean theorem, ...
c² = a² +b²
Substituting for c², we have ...
sin²(θ) +cos²(θ) = (a² +b²)/(a² +b²)
sin²(θ) +cos²(θ) = 1
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Additional comment
This trig identity also goes by the name "Pythagorean identity."
