Verify that each equation is an identity.
2 cos³ x - cos...

Question

Verify that each equation is an identity.

2 cos³ x - cos x = (cos² x - sin² x)/sec x

Accepted Answer

Hey, everyone here we are asked is the given equation and identity. Here we are given negative two sine C of X plus sine of X is equal to cosine of two X divided by cos of X. We have two answer choice options, answer a yes and answer B no. In order to determine if this equation is an identity, we need to verify that the left hand side is equivalent to the right hand side. So that means we want to take a closer look at the left hand side of the given equation. Again, recalling that we have negative two sine cubed of X plus sine of X. So now we want to try to manipulate this expression so that it matches the right hand side. And so one thing we can notice here is that both of the terms contain sin of X, which means that we can simply factor out sin of X as our first step. So factoring it out, we have s of X multiplied by the quantity of one minus two S squared of X. And so now looking back at the right hand side of our given equation, we can notice that we have coin of X in the denominator. And so we can make our current expression, have coin of X in the denominator as well by recalling that sine of X is equivalent to one divided by cosin of X. And so now just substituting in this expression for sin of X, we can rewrite our current expression as the quantity of one minus two sine squared of X divided by coin of X. And so now we are one step closer to matching the right hand side. And so the next thing we want to notice is that the expression in the numerator of our current expression that we're working with matches the double angle formula for cosine. So we need to recall that cosine of two X is equivalent to one minus two sine squared of X. And now with this formula identified, we know we can substitute in cosine of two X into our numerator. So doing this, we have cosine of two X divided by coin of X. Now that we have fully manipulated our left hand side of the equation, we can look back at the right hand side of the given equation. And we notice that they are equivalent to each other or the left hand side is in fact equivalent to to the right hand side. Therefore, we can conclude that yes, our given equation is in fact an identity. Therefore, we are left with answer choice. A where our answer here is. Yes. 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.
2 cos³ x - cos x = (cos² x - sin² x)/sec x

Key Concepts

Trigonometric Identities

Secant Function

Factoring and Simplifying Expressions

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