Show that
sinx = (square root of) 1 - cos^2x
is not an identity
All we need is one counterexample to show it is not an identity.
let x = 210º
LS = sin 210 = -1/2
RS = √(1- cos^2 (210º))
= √(.25)
= 1/2
So there!
Well, let's put on our mathematical clown shoes and dive into this question!
To show that the equation sin(x) = √(1 - cos^2(x)) is not an identity, we need to find a value of x for which the equation doesn't hold.
Let's start by squaring both sides of the equation:
sin^2(x) = 1 - cos^2(x)
Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can rewrite the equation as:
1 - cos^2(x) = 1 - cos^2(x)
As you can see, we get the exact same expression on both sides of the equation, which means that the equation is indeed an identity.
So, it seems like I wasn't able to find a way to disprove it. Looks like mathematics has won this round, and my clownish antics didn't get very far. But no worries, there are plenty more humorous math questions to tackle!
To show that an equation is not an identity, we need to find at least one value of x for which the equation is not satisfied. Let's solve the equation and see if it holds true for all values.
Starting with the equation:
sinx = √(1 - cos^2x)
Square both sides of the equation:
sin^2x = 1 - cos^2x
Rearrange the equation by moving the cos^2x to the left side:
sin^2x + cos^2x = 1
Since sin^2x + cos^2x is the identity for all values of x (trigonometric identity), this equation is always satisfied. Therefore, the given equation is indeed an identity.
To show that the equation:
sin(x) = sqrt(1 - cos^2(x))
is not an identity, we can simplify both sides of the equation and check if they match for all values of x. If there is even a single value of x for which the equation does not hold, then it is not an identity.
First, let's square both sides of the equation:
sin^2(x) = 1 - cos^2(x)
Using the identity sin^2(x) + cos^2(x) = 1, we can substitute sin^2(x) with (1 - cos^2(x)):
1 - cos^2(x) = 1 - cos^2(x)
Both sides of the equation are equal, which means that the original equation sin(x) = sqrt(1 - cos^2(x)) is actually an identity.
Therefore, the equation sin(x) = sqrt(1 - cos^2(x)) is indeed an identity.