How do you verify $tan (x+pi/2)= -cot x$ ?

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Answer 1 · 2016-02-16T18:46:55

By applying the basic trigonometric relations. see below

head

Key-relation 1. $tanx=sinx/cosx$

Key-relation 2. $cotx=1/tanx=cosx/sinx$

Key-relation 3. $cos(a+b)=cosa*cosb -sina*sinb$

Key-relation 4. $sin(a+b)=cosa*sinb + sina*cosb$

Import results

Important result 1. $cos (pi/2)=0$
Important result 1. $sin (pi/2)=1$

Gathering

By using all the knowledge left so far and some mathematical tricks, we have:

$sin(x+(pi/2))=cosx*sin(pi/2) + sinx*cos(pi/2)$
$sin(x+(pi/2))=cosx$

Further:

$cos(x+(pi/2))=cosx*cos(pi/2) - sinx*sin(pi/2)$

$cos(x+(pi/2))=- sinx$

Finally:

$tan(x+(pi/2))=sin(x+(pi/2))/cos(x+(pi/2))$

$tan(x+(pi/2))=-cosx/sinx=cotx$

End of the proof!

How do you verify tan (x+pi/2)= -cot x tan (x+pi/2)= -cot x?