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

Question

Verify that each equation is an identity.

(sin 2x)/(sin x) = 2/sec x

Accepted Answer

Hey, everyone here we are asked is the given equation and identity. Here we have 10 sine of two X divided by cosine of X is equal to 20 divided by coin of X. We have two answer, choice options, answer a yes and answer B no. In order to determine if the equation is, in fact 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 our left hand side recalling that we have 10 sine of two X divided by cosine of X. And so now we want to manipulate the left hand side as much as possible to try and make it look like the right hand side. So one of the first things we can notice is in the numerator, we have sine of two X. This tells us we need to recall the double ankle formula for a sine function which states that sine of two X is equivalent to two multiplied by sine of X multiplied by cosine of X. And so now we can substitute in this expanded expression into the numerator. And so we now have 10 multiplied by two, multiplied by sine of X multiplied by cosine of X. And this is all still divided by cosine of X. And now we just want to start simplifying. So first we notice we can multiply together 10 and two. And we also notice that cosine in the denominator will cancel with the cosine in the numerator. And so we have just 20 multiplied by sine of X. And now the last thing we can do to rearrange our left hand side of the equation is to recall that sine of X is equivalent to one divided by cosecant of X. And so now just substituting in our new expression for sin of X, our final expression for the left hand side becomes 20 divided by cosecant of X. And now we can refer back to our given equation looking at the right hand side, we notice that our current left hand side is exactly the same or equivalent to this right hand side given to us. And because the left hand side is equivalent to the right hand side, we can conclude that yes, our 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.
(sin 2x)/(sin x) = 2/sec x

Key Concepts

Trigonometric Identities

Double-Angle Formulas

Reciprocal Trigonometric Functions

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