Verify that each equation is an identity.
(sin 3t + sin ...

Question

Verify that each equation is an identity.

(sin 3t + sin 2t)/(sin 3t - sin 2t ) = tan (5t/2)/(tan (t/2))

Accepted Answer

Hey, everyone in this problem, we're asked if the given equation is an identity. And when we're asked that question, when it's really asking is, does this equation hold true for every X be? The equation were given sine of seven X plus sine of six X, all divided by sine of seven X minus sine of six X is equal to a tangent of 13 X divided by two divided by tangent of X divided by two. We have to answer choices. Option A yes. Option B no, let's start with the left hand side in this case. OK. So we have sign of seven X plus sign of six X divided by sine of seven X minus sine A six X. And we wanna see if we can write this as the right hand side of this equation. OK. So we're gonna use some identities and try to simplify this. And on the right hand side, we can see that everything is multiplied or divided together on the left hand side that we're dealing with things are added, things are subtracted. And so what we're gonna use is our sum to product formula and recall that our sum to product formula tells us that if we have sign of A, what sign of B, we can write this as two multiplied by sign of A plus B divided by two multiplied by cosine of A minus B divided by two. Similarly, if we have sign of A minus xy, of fee, we can write this as two cosine of A plus B divided by two multiplied by a sign of A minus B divided by two. OK. So we have these two sums of product formulas. One for the addition, one for the subtraction. OK. What we have in the numerator is the addition. What we have in the denominator is the subtraction. OK? Where we have seven X being equal to A and six X being equal to B. If we apply this formula, we can simplify our expression and write it as two multiplied by sign of seven X plus six X all divided by two, multiplied by cosine of seven X minus six X all divided by two. All of this is gonna be divided by two sign. Oops cosine two cosine seven X 46 X divided by two multiplied by S seven X minus six X divided by two. Let's simplify inside of our trip functions. So we have two multiplied by sign of 13 X divided by two multiplied by cosine of X divided by two, divided by two cosine of 13 X divided by two multiplied by sine of X divided by two. And now what do we have? Well, the twos will divide oh, we have sine 13 X divided by two, divided by cosine of 13 X divided by two sine divided by cosine is equivalent to trig or sorry to tan. And so we end up with tangent of 13 X divided by two. And then we're still multiplying by cosine of X divided by two divided by sine of X divided by two cosine divided by sine. That's cotangent or one divided by tangent. We're gonna start by writing it as cotangent of X divided by two. But to get this to look like the right hand side, we know we can write this as one divided by tangent. And so our expression becomes tangent of 13 X divided by two divided by tangent of X divided by two. And this is equal to the right hand side of our equation. OK. So the left hand side is equal to the right hand side. It doesn't matter what X value we substitute. And we've just shown that these two are equal. And so this is an identity going to our answer choices. That means that the correct answer is going to be option A yes. Thanks everyone for watching. I hope this video see you in the next one.

Verify that each equation is an identity.
(sin 3t + sin 2t)/(sin 3t - sin 2t ) = tan (5t/2)/(tan (t/2))

Key Concepts

Trigonometric Identities

Sum-to-Product Formulas

Tangent Function and Angle Relationships

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