In Exercises 1–60, verify each identity.

sin x sec x = tan x

Question

In Exercises 1–60, verify each identity.

sin x sec x = tan x

Accepted Answer

Welcome back, everyone. We want to determine whether the given identity is true or false. The identity says that the coi kind of X multiplied by the core tangent of X multiplied by the cosine of X is equal to the square of the cotangent of X. For our answer choices A says it's true. While B says it's false. Now to determine the truth value of the given identity, we can simplify both sides and see if they are equal. Notice that the left hand side looks a bit more complicated than the right. So let's start there to help us figure this out. Let's think about all the definitions for our trigonometric functions for trigonometric identities recall that we have three trigonometric functions along with the reciprocal. So we know that the sine of XK is equal to one divided by the cosecant of X. OK. In other words, the cosecant is the reciprocal of the sine function. Likewise, we also know that the cosine function is the reciprocal of the second. OK. So the core, the cosine of X equals one divided by the S kind of X. And we also know that the tangent, the tangent is actually the ratio of the sine function to the cosine function. And since the core tangent is the reciprocal of the tangent, that means it's going to be equal to the cosine of a function divided by the sine of the function. OK. And now with what we're working with, you can realize we'd really be interested then in the core tangent, the sine function and the cosine function. Now I'm going to use the cosecant here because we're interested in san because it's connected to the cosecant. OK. So since the core second is the reciprocal of the sine function, that means the cos can equals one divided by sine X. So starting with the left hand side, OK, then we can write our cos of X as one divided by the sine of X. We're going to multiply that by the core tangent of X, which is the cosine of X divided by the sine of X. And we're also going to multiply that by the cosine of X. OK. Now here notice that we have a cosine of X multiplying the cosine of X. So that's the square of the cosine of X divided by the sine of X multiplied by the sine of X which is the square of the sine of X. We can write this even further, OK? To say that this is the cosine of X divided by the sine of X or squared. Now, this is pretty interesting because remember earlier, we defined the core tangent as the cosine of the core tangent of X as the cosine of X divided by the sine of X. Therefore, we can replace the core sine of X divided by the sine of X with the core tangent. So it will be equal to the square of the core, the core tangent of X. Therefore, we've proved that the left hand side of our equation will eventually be equal to the right hand side of our equation. Therefore, the given identity is true. A is the correct answer. Thanks for watching everyone. I hope this video helped.

In Exercises 1–60, verify each identity. sin x sec x = tan x

Key Concepts

Trigonometric Identities

Reciprocal Functions

Quotient Identity

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