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=== Legendre’s identity ===
Legendre proved the following identity:
:<math display="block">K(\cos \theta) E(\sin \theta ) + K(\sin \theta ) E(\cos \theta) - K(\cos \theta) K(\sin \theta) = {\pi \over 2}, \text{ for all } \theta. </math><ref name="brent" />
=== Elementary proof with integral calculus ===
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