Verify that each equation is an identity.
sin³ θ + cos³ ...

Question

Verify that each equation is an identity.

sin³ θ + cos³ θ = (cos θ + sin θ) (1 - cos θ sin θ)

Accepted Answer

Hey, everyone here we are asked is the given equation and identity. We are given sine cubed of X minus cosine cubed of X is equal to the quantity of sine of X minus cosine of X multiplied by the quantity of one plus sine of X multiplied by cosine of X. We have two answer, choice options, answer A yes and answer B no. So in order to determine if our equation is an identity, we need to verify that the left hand side is equivalent to the right hand side. So taking a closer look at our left hand side, we recall we have sine cubed of X minus cosine cubed of X. And so we want to start by fully expanding this expression. And we can do this by recalling the difference of cubes formula which states that A cubed minus B cubed is equivalent to the quantity of A minus B multiplied by the quantity of A squared plus B squared plus A multiplied by B. So now applying this formula to our current expression, we can expand it. So again, we have sine cubed of X minus cosine cubed of X. And we now know that this is equivalent to the of sine of X minus cosine of X multiplied by the quantity of sine squared of X plus cosine squared of X plus sine of X multiplied by cosine of X. Now, looking at our current expression, we can notice we have sine squared of X plus cosine squared of X. This follows the pythagorean identity, which if we recall, we have sine squared of X plus cosine squared of X is equivalent to one. So now we can simply substitute in one for this expression. And so now rewriting our current expression becomes the quantity of sine of X minus cosine of X multiplied by the quantity of one plus sine of X multiplied by cosine of X. And now we can look back at our original equation and we can look at the right hand side and we will notice that our left hand side is in fact equivalent to the right hand side. And since they are equivalent to each other, we can conclude that yes, our given equation is in fact an identity. And with this, we are left with answer choice. A where again the answer is yes. The given equation is an identity. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Verify that each equation is an identity.
sin³ θ + cos³ θ = (cos θ + sin θ) (1 - cos θ sin θ)

Key Concepts

Trigonometric Identities

Algebraic Factorization

Pythagorean Identity

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