A.Verify the identity.
B.Determine if the identity is true for the given value of x. Explain.
[cot(x) – 1] / [cot(x) + 1] = [1 – tan(x)] / [1 + tan(x)], x = π/4
I solved A and I believe it's true; however, I need help with B.
Put in for x the value.
[cot(x) – 1] / [cot(x) + 1] = [1 – tan(x)] / [1 + tan(x)]
or
[cot(PI/4) – 1] / [cot(PI/4) + 1] = [1 – tan(PI/4)] / [1 + tan(PI/4)]
or
(1-1)/2=?=(1-1)/(1+1)
0=0
To determine if the identity is true for the given value of x, which is x = π/4, we need to substitute the value of x into both sides of the equation and simplify.
Let's start with the left side of the equation:
[cot(x) – 1] / [cot(x) + 1]
Substituting x = π/4:
[cot(π/4) – 1] / [cot(π/4) + 1]
Now, let's find the cotangent of π/4. The cotangent is the reciprocal of the tangent function.
cot(π/4) = 1 / tan(π/4)
We know that tan(π/4) = 1, so we can substitute this value:
cot(π/4) = 1 / 1 = 1
Now we can substitute this value back into the equation:
[(1) – 1] / [(1) + 1]
Simplifying further:
[0] / [2] = 0
So, the left side of the equation is 0 when x = π/4.
Now let's evaluate the right side of the equation:
[1 – tan(x)] / [1 + tan(x)]
Substituting x = π/4:
[1 – tan(π/4)] / [1 + tan(π/4)]
Since we already know tan(π/4) = 1:
[1 – 1] / [1 + 1]
[0] / [2] = 0
So, the right side of the equation is also 0 when x = π/4.
Since both the left side and the right side of the equation evaluate to 0 when x = π/4, we can conclude that the identity is true for this particular value of x.