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

Question

12590 views around the world

You can reuse this answer
Creative Commons License

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

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

$square$

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