In this discussion, we prove that a Trig Equation is NOT an Identity.
Here is an outline of this discussion:
🔹Proving sin(x) = 1-cos(x) is Not an Identity
🔹Proving cosx = sqrt(1-sin^2x) is Not an Identity
Now when proving identities, always begin with one side of an equation and slowly manipulate it until it looks like the other side.
However, when disproving identities, all you have to do is find one value that plugs into the equation such that one side of the equation does not equal the other side.
Note that, just because a few values make the equation true does not mean that the equation is an identity.
You have to keep playing around with numbers until you find one that shows the equation is false.
Let us begin!
🔹Proving sin(x) = 1-cos(x) is Not an Identity
For the first example, we have:
sin(x) = 1-cos(x).
We want to show that the above equation is false.
For the first guess, choose an angle in quadrant 1.
Let x= pi/4.
Clearly, the left hand side does not equal the right hand side.
Thus, sin(x) = 1-cos(x) is not an identity.
🔹Proving cosx = sqrt(1-sin^2x) is Not an Identity
For the next example, we have:
cosx = sqrt(1-sin^2x) .
Now this one is a little tricky because one might think that squaring both sides of the equation will prove the identity.
Squaring both sides, we have
cos^2x = 1-sin^2x.
Now, cos^2x = 1-sin^2x IS a Pythagorean identity.
However, it doesn’t mean that cosx = sqrt(1-sin^2x) is an identity because, anytime you square both sides of an equation, you may get a solution that does not agree with the original problem.
For example, suppose z= 4.
Squaring both sides, we have z^2 =16.
Now solving for z we have z= +-4.
Notice that you have two values for z after squaring both sides, but you only had one value for z in the original problem.
That is why finding a value for x such that cosx = sqrt(1-sin^2x) is a false statement is the way to go.
For the first choice, let x=pi/4
We see that the left side of the equation equals the right side.
So we need to choose another value.
For the second choice, let x= 3pi/4.
We see that the left hand side of the equation does not equal the right hand side.
Thus, cosx = sqrt(1-sin^2x) is not an identity.
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