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.