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

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Answer 1 · 2015-06-09T15:45:13

Verify tan (x + pi/2) = - cot x

On the trig unit circle, the value AT = tan x rotates counterclockwise an arc of pi/2, and becomes BU = - cot x

Answer 2 · 2016-06-24T20:15:02

Note that #tan(x+pi/2)=sin(x+pi/2)/cos(x+pi/2)#.

For the numerator, use #sin(A+B)=sin(A)cos(B)+cos(A)sin(B)#:

#sin(x+pi/2)=sin(x)cos(pi/2)+cos(x)sin(pi/2)#

#=sin(x)*0+cos(x)*1#

#=cos(x)#

In the denominator, use #cos(A+B)=cos(A)cos(B)-sin(A)sin(B)#:

#cos(x+pi/2)=cos(x)cos(pi/2)-sin(x)sin(pi/2)#

#=cos(x)*0-sin(x)*1#

#=-sin(x)#

Thus, we see that

#tan(x+pi/2)=sin(x+pi/2)/cos(x+pi/2)=cos(x)/(-sin(x))=-cot(x)#

#square#