A.Verify the identity.
B.Determine if the identity is true for the given value of x. Explain.
Sec(x) / tan(x) = tan(x) / [tan(x) – cos(x)],x = π
I already solved A and I believe it's true; however, I need help with B.
clearly B is false at x=π
Sec(x) / tan(x) = tan(x) / [tan(x) – cos(x)]
(-1)/0 = 0/(0-(-1)) = 0/1
-∞ = 0
I don't think so!
To determine if the identity is true for the given value of x = π in the equation sec(x) / tan(x) = tan(x) / [tan(x) - cos(x)], we need to substitute the value of x and simplify on both sides.
Let's start with the left-hand side of the equation:
sec(x) / tan(x) = sec(π) / tan(π)
Secant function (sec) is defined as 1/cos(x), and tangent function (tan) is defined as sin(x)/cos(x).
Substituting the values for π:
sec(π) = 1/cos(π)
Since cos(π) = -1, we have:
sec(π) = 1/(-1) = -1
tan(π) = sin(π)/cos(π)
Since sin(π) = 0 and cos(π) = -1, we have:
tan(π) = 0/(-1) = 0
Now, let's simplify the right-hand side of the equation:
tan(x) / [tan(x) - cos(x)] = tan(π) / [tan(π) - cos(π)]
Using the values we determined earlier:
tan(π) = 0
cos(π) = -1
Substituting:
tan(π) / [tan(π) - cos(π)] = 0 / [0 - (-1)]
Simplifying the denominator:
0 - (-1) = 0 + 1 = 1
So, the right-hand side becomes:
tan(π) / [tan(π) - cos(π)] = 0 / 1 = 0
Comparing the left-hand side and the right-hand side:
sec(π) / tan(π) = -1 / 0 ≠ 0
Therefore, the identity is not true for the given value of x = π.
To determine if the given identity is true for the given value of x = π, we need to substitute π into both sides of the equation and check if the equation remains true.
Let's start by evaluating the left-hand side (LHS) of the equation:
LHS = sec(x) / tan(x)
Substituting x = π, we have:
LHS = sec(π) / tan(π)
To proceed, we need to recall the trigonometric definitions for secant and tangent. The secant of an angle is the reciprocal of the cosine, and the tangent of an angle is the sine divided by the cosine. Using these definitions, we can express sec(π) and tan(π) as follows:
sec(π) = 1 / cos(π)
tan(π) = sin(π) / cos(π)
Recall that the cosine of π is -1, and the sine of π is 0. Substituting these values, we get:
sec(π) = 1 / (-1) = -1
tan(π) = 0 / (-1) = 0
Now, let's substitute these values back into the LHS:
LHS = (-1) / 0
At this point, we can see that the LHS is undefined since we cannot divide by zero. Therefore, the given identity is not true for the value of x = π.
In summary, when x = π, the given identity is not true since the left-hand side is undefined.