Which of the following values for x is a counterexample to the claim that cot(pi/2-x)=tan x is an identity?
Question options:
a)
pi
b)
pi/2
c)
cot(pi/2-x)=tan x is an identity
d)
3pi/4
C - always true since
co-f(π/2 - x) = f(x)
The co- in co-sin, co-tan, co-sec means "complement of x"
Yes, that's correct! The identity cot(pi/2-x) = tan x is always true.
To determine which of the given values for x is a counterexample to the claim that cot(pi/2-x) = tan x is an identity, we need to substitute each value into the equation and check if the equation holds true.
The identity cot(pi/2-x) = tan x holds true for all values of x except for the values where the tangent function is undefined, which occur when x is equal to pi/2 + k*pi, where k is an integer.
Let's substitute each value of x into the equation and see if it holds true:
a) For x = pi:
cot(pi/2-x) = cot(pi/2-pi) = cot(-pi/2) = 0 (since cotangent is undefined at -pi/2)
tan x = tan(pi) = 0
Since both sides of the equation are equal to 0, the equation holds true for x = pi. Therefore, it is not a counterexample.
b) For x = pi/2:
cot(pi/2-x) = cot(pi/2-pi/2) = cot(0) = undefined (since cotangent is undefined at 0)
tan x = tan(pi/2) = undefined
Both sides of the equation are undefined for x = pi/2, so it does not provide a counterexample.
c) Since this option states that cot(pi/2-x) = tan x is an identity, it is not a specific value of x and therefore cannot be a counterexample.
d) For x = 3pi/4:
cot(pi/2-x) = cot(pi/2-3pi/4) = cot(-pi/4) = -1 (cot(-pi/4) = -1)
tan x = tan(3pi/4) = -1
Both sides of the equation are equal to -1 for x = 3pi/4, so it does not provide a counterexample either.
Therefore, none of the given values for x is a counterexample to the claim that cot(pi/2-x) = tan x is an identity.