Verify that each equation is an identity.
(2 cot x)/(tan...

Question

Verify that each equation is an identity.

(2 cot x)/(tan 2x) = csc² x - 2

Accepted Answer

Hey, everyone here we are is the given equation and identity. Here we are given two sine of X divided by tangent of two X is equal to two cosine of X minus secret of X. Here we have two answer choice options, answer a yes and answer B no. In order to determine if our equation is, in fact an identity, we need to verify that the left hand side is equivalent to the right hand side. So looking at the left hand side, we recall, we are given two sign of X divided by tangent of two X. And now we want to manipulate the left hand side so that we can try to match the right hand side of the given equation. And so the first thing we can do is expand the denominator. We notice we have tangent of two X which tells us we can apply the double angle formula for the tangent function, which if we recall this formula, we know that tangent of two X is equivalent to two multiplied by tangent of X divided by the quantity of one minus tangent squared of X. And now we can just substitute in this expression into our denominator. And so our left hand side of the equation becomes two sine of X divided by two tangent of X divided by the quantity of one minus tangent squared of X. And now to make this expression easier to simplify, we see we are dividing by a fraction. Well, we can instead rewrite it as multiplying by its reciprocal. And so we have two sign of X multiplied by the quantity of one minus tangent squared of X divided by two tangent of X. Now, we can notice that we have a two in the numerator and the denominator. So they cancel each other out. And now continuing to manipulate this current expression, we can now manipulate the denominator here where we have tangent of X by recalling that tangent of X is equivalent to sine of X divided by cosine of X. So now just substituting in this expression into the denominator, our current expression becomes sine of X multiplied by the quantity of one minus tangent squared of X divided by sine of X divided by cosine of X. And so now we want to start simplifying once again. And we can notice here that the sign on the left side of this expression will cancel with the sign in the denominator. And so now just rewriting, we have cosine of X multiplied by the quantity of one minus tangent squared of X. And now we want to substitute in the expression for tangent of X, which we previously identified into our current expression. And so doing this, we have cosine of X multiplied by the quantity of one minus sine squared of X divided by cosine squared of X. And now just distributing in cosine of X into our expression. Here in the parentheses, we get cosine of X minus sine squared of X divided by cosine of X. Now, if we recall the pythagorean identity, we know that sine squared of X is equivalent to one minus cosine squared of X. And so now we can just substitute in this new expression into the, the numerator of our current expression. And so we have cosine of X minus the quantity of one minus cosine squared of X all divided by cosine of X. And now our next step is to just expand and simplify this fraction here. And so we have cosine of X minus the quantity of one divided by cosine of X minus cosine of X. And now we just need to recall that one divided by cosine of X is equivalent to second of X. And with this, we can continue to simplify our expression here. So rewriting, we have cosine of X minus the quantity of second of X minus cosine of X. And now just distributing the negative, we have cosine of X minus sin of X plus cosine of X. And now just combining like terms, we are left with two cosine of X minus sin. Of X. And now comparing our final expression for the left hand side to our original equations, right hand side, we can notice that the left hand side is in fact equivalent to the right hand side. So this tells us that 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, our final answer is yes, our 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.
(2 cot x)/(tan 2x) = csc² x - 2

Key Concepts

Trigonometric Identities

Double-Angle Formulas

Reciprocal and Quotient Identities

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