Lemma: tan π/4 = 1
Proof:
(1) Using definition:
tan(π/4) = sin(π/4)/cos(π/4)
(2) Using the triangle definition of sin and cosine (see here), we know that:
sin (x) = cos (π/2 - x)
(3) This then gives us that:
sin (π/4) = cos(π/2 - π/4) = cos(2π/4 - π/4) = cos(π/4)
(4) So that:
tan(π/4) = cos(π/4)/cos(π/4) = 1
QED
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