How do you verify the identity: tan (x + pi/2) = -cot xtan(x+π2)=−cotx?
2 Answers
Verify tan (x + pi/2) = - cot x
Explanation:
On the trig unit circle, the value AT = tan x rotates counterclockwise an arc of pi/2, and becomes BU = - cot x
Note that
For the numerator, use
sin(x+pi/2)=sin(x)cos(pi/2)+cos(x)sin(pi/2)sin(x+π2)=sin(x)cos(π2)+cos(x)sin(π2)
=sin(x)*0+cos(x)*1=sin(x)⋅0+cos(x)⋅1
=cos(x)=cos(x)
In the denominator, use
cos(x+pi/2)=cos(x)cos(pi/2)-sin(x)sin(pi/2)cos(x+π2)=cos(x)cos(π2)−sin(x)sin(π2)
=cos(x)*0-sin(x)*1=cos(x)⋅0−sin(x)⋅1
=-sin(x)=−sin(x)
Thus, we see that
tan(x+pi/2)=sin(x+pi/2)/cos(x+pi/2)=cos(x)/(-sin(x))=-cot(x)tan(x+π2)=sin(x+π2)cos(x+π2)=cos(x)−sin(x)=−cot(x)
square
