Find the counterexample to shows that the equation sec (alpha) - cos (alpha) = sin (alpha) sec (alpha) is not an identity.
Please explain.
To find a counterexample, we need to find a specific value for α (alpha) that makes the equation false. Let's proceed step by step.
First, let's assume the equation sec(α) - cos(α) = sin(α) sec(α) is an identity. This means it should hold true for any value of α.
Now, we'll plug in a specific value of α and see if the equation holds.
Let's try α = 0:
sec(0) - cos(0) = sin(0) sec(0)
Using the definitions of secant, cosine, and sine:
1 - 1 = 0 * 1
Simplifying further:
0 = 0
As we can see, the equation holds true for α = 0.
However, to show that the equation is not an identity, we only need to find one counterexample where the equation does not hold.
Let's try another value of α, for example, α = π/2:
sec(π/2) - cos(π/2) = sin(π/2) sec(π/2)
Using the definitions of secant, cosine, and sine:
1 - 0 = 1 * 1
Simplifying further:
1 = 1
As we can see, the equation also holds true for α = π/2.
Therefore, from these examples, it might suggest that the equation is indeed an identity. However, before we conclude, let's check one more value of α.
Let's try α = π:
sec(π) - cos(π) = sin(π) sec(π)
Using the definitions of secant, cosine, and sine:
-1 - (-1) = 0 * (-1)
Simplifying further:
0 = 0
As we can see, the equation holds true for α = π.
Since we found counterexamples where the equation holds true, it means the equation sec(α) - cos(α) = sin(α) sec(α) is not an identity.
there are many counterexamples. For instance,
sec(pi/4) - cos(pi/4) = √2 - 1/√2
sin(pi/4)*sec(pi/4) = 1/√2 * √2 = 1
not equal. In fact, for any alpha not a multiple of pi, they are not equal.