Hi, could someone explain to me how to use a counterexample in trig to prove whether something is either an identity or equation?

For ex. how would you do that with this question?

sec^2x-tan^2x= cos x

A counterexample is simply any instance that disproves the statement.

For example, try using x = pi/4. Note that this is NOT the only correct answer.

Would I substitute the same number to both sides?

Certainly! To use a counterexample to prove whether something is an identity or equation in trigonometry, you need to find a specific value or set of values that satisfy one side of the equation, but not the other.

Let's consider the equation you mentioned: sec^2x - tan^2x = cos x.

To determine if this equation holds true for all values of x (i.e., it is an identity), we need to show that both sides of the equation are equivalent.

First, let's simplify each side of the equation using trigonometric identities.

The left side can be simplified using the Pythagorean identity, which states that sec^2x - tan^2x = 1:

1 - tan^2x = cos x.

Now, we need to check if this simplified equation holds true for all values of x.

To use a counterexample, we need to find a value of x that satisfies one side of the equation but not the other. In this case, we can choose x = 𝜋/4.

For x = 𝜋/4, the left side of the equation becomes:

1 - tan^2(𝜋/4) = 1 - 1 = 0.

However, the right side of the equation becomes:

cos(𝜋/4) = 1/√2.

Since 0 and 1/√2 are not equal, we have found a counterexample that disproves the original equation. Therefore, sec^2x - tan^2x = cos x is not an identity.

Using a counterexample, we have shown that the equation is not true for all values of x and, therefore, is not an identity.

I hope this helps clarify how to use a counterexample to prove whether something is an identity or equation in trigonometry!