Verify the identity cot{0-pi/2=-tan 0
use the identities:
tan(x)=sin(x)/cos(x)
cot(x)=cos(x)/sin(x)
and
sin(π/2)=1, cos(π/2)=0
sin(0)=0, cos(0)=1
I think she means θ when using 0
I'll use x to avoid copy/paste of special characters.
one way:
recall the definition of co-functions: function of complementary angle. So,
cos(x) = sin(pi/2 - x)
cot(x) = tan(pi/2 -x)
so, cot(x-pi/2) = -cot(pi/2 - x) = -tan(x)
or, use cot addition formula:
cot(x-pi/2) = (1+cotx cot pi/2)/(cot pi/2 - cot x) = (1+0)/(0-cotx) = 1/-cotx = -tanx
Thanks Steve, good thought!
To verify the given trigonometric identity cot(0 - π/2) = -tan(0), we can use the definitions of cotangent and tangent.
The cotangent function can be defined as the reciprocal of the tangent function: cot(x) = 1/tan(x).
Let's evaluate the left side of the identity, cot(0 - π/2):
1. Substitute x = 0 - π/2 into the cotangent function: cot(0 - π/2) = 1/tan(0 - π/2).
2. Recall that tan(x - π/2) = -cot(x), which means that tan(0 - π/2) = -cot(0):
cot(0 - π/2) = 1/(-cot(0)).
3. Rewrite cot(0) as 1/tan(0):
cot(0 - π/2) = 1/(-(1/tan(0))) = -tan(0).
Therefore, the left side of the identity, cot(0 - π/2), is equal to the right side, -tan(0). Hence, the given identity is verified.