Verify that each equation is an identity.
sin³ θ = sin θ...

Question

Verify that each equation is an identity.

sin³ θ = sin θ - cos² θ sin θ

Accepted Answer

Hey, everyone in this problem, we're asked if the given equation is an identity. When we say identity here. What that means is that, that equation holds true for any X value. The equation we're given cosine cubed of X is equal to sine of X minus sine squared of X cosine of X. We have two answer choices. Option A yes and option B no. So what we're gonna do is start with the right hand side of the equation in this case. OK. It's a little bit more complicated, more expanded version. So we're gonna try to simplify it down and see if we can get that left hand side. We're gonna start, we have sin of X minus sine squared of X multiplied by cosines. Now, we want to get this in terms of cosine of X as much as we can because that left hand side is cosine cubed of X. Anything that we can relate to cosine, we're going to and sine squared of X is something that we can do. OK? Sine squared of X plus cosine squared of X is equal to one. OK. Recall that identity. What that means is that Sine squared of X is going to be equal to one minus cosine squared of X. And just rearranging that equation subtracting cosine squared of X from both sides. OK. So if we do that, then the right hand side of our equation is going to become sin of X minus one minus cosine squared of X multiplied by cosine effects. Let's expand and simplify. We get side effects minus cosine of X plus cosine QX. And what we can tell is that this is not going to be equal to cosine cubed of X. OK. We have this cosine cubed X term which is great, but we also have sine of X minus cosine sine of X minus cosine of X is not going to be equal to zero for every X value. And so this right hand side of our equation won't be equal to cosine cubed of X. And so this is not equal to the left hand side of our equation. And therefore this is not an identity. And so the correct answer choice here is option B No, thanks everyone for watching. I hope this video helped you in the next one.

Verify that each equation is an identity.
sin³ θ = sin θ - cos² θ sin θ

Key Concepts

Trigonometric Identities

Pythagorean Identity

Algebraic Manipulation of Trigonometric Expressions

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