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(1) tan(x+90) = -cot(x)
Use the trig identity
(2) tan(x) = sin(x)/cos(x) to get
(3) tan(x+90) = sin(x+90)/cos(x+90)
Now use the trig identities
(4) sin(a+b) = sin(a)cos(b) + cos(a)sin(b) and
(5) cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
in (3) to get
(6) tan(x+90) = (sin(x)cos(90) + cos(x)sin(90))/(cos(x)cos(90) - sin(x)sin(90))
Now using
(7) cos(90) = 0
(8) sin(90) = 1
in (6) we get
(9) tan(x+90) = (sin(x)*0 + cos(x)*1)/(cos(x)*0 - sin(x)*1) or
(10) tan(x+90) = cos(x)/(-sin(x)) or
(11) tan(x+90) = -cos(x)/sin(x)
Use the trig identity
(12) cot(a) = cos(a)/sin(a) (11) becomes
(13) tan(x+90) = -cot(x) QED
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