Verify that each equation is an identity.
(2 tan B)/(sin...

Question

Verify that each equation is an identity.

(2 tan B)/(sin 2B) = sec² B

Accepted Answer

Hey, everyone here we are asked is the given equation and identity. Here we are given four cotangent of X divided by sine of two X is equal to two cos squared of X. We have two answer, choice options, answer a yes and answer B no. To determine if our given 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 our given equation where as we recall, we have four code tangent of X divided by sine of two X. And so now our goal here is to manipulate this expression so that it matches the right hand side of the given equation. So one of the first things we can do is look at the denominator and notice we have sine of two X. Well, this tells us we need to recall the double angle formula for the sine function which states that sine of two X is equivalent to two sine of X multiplied by cosine of X. So now we can substitute in this new expression into the denominator And so we have four cotangent of X divided by two sine of X multiplied by cosine of X. And now we want to rewrite our numerator so that it also contains sine or cosine. And so we need to recall that code tangent of X is equivalent to cosine of X divided by sine of X. And now just substituting in this expression, we have four cosine of X divided by sine of X divided by two sine of X multiplied by cosine of X. And now to make this expression easier to simplify, we can rewrite it as four cosine of X divided by sine of X multiplied by one divided by two sine of X multiplied by cosine of X. And now we can notice that we have a cosine in the numerator and the denominator which will cancel each other out. We also have a four in the numerator and a two in the denominator. So the four in the numerator becomes a two. And now just rewriting our simplified expression, we have two divided by sine squared of X. And now our last step is to recall that coin of X is equivalent to one divided by sine of X. And so now applying this expression to our current expression, we have now two multiplied by cos squared of X. And with this, we have fully rearranged the left hand side of our equation. And so we can refer back to our given equation, its right hand side. And we noticed that the left hand side is in fact equivalent to the right hand side. And from this, we can conclude that yes, the given equation is in fact an identity. Therefore, we are left with answer choice. A where our final answer 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 tan B)/(sin 2B) = sec² B

Key Concepts

Trigonometric Identities

Double-Angle Formulas

Basic Trigonometric Ratios and Reciprocal Identities

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