In Exercises 1–60, verify each identity.

tan (-x) cos x = -sin x

Question

In Exercises 1–60, verify each identity.

tan (-x) cos x = -sin x

Accepted Answer

Hello, everybody. I hope you're doing all right. Today, today we're going to be looking at this question that states determine whether the given identity is true or false where the given identity is cotangent of open parentheses, negative acts closed parentheses multiplied by cosine of X is equal to sin of X minus co C can ox. Now we only have two answer choices. In this case, a being true and B being false. So for a question like this, I'm going to be tackling each side of the equation first. So since that prove that they're either true or false, I'm going to be solving each side of the equation one by one and then see if they match up at the end. So first, let's talk about the left hand side of the equation. So everything I'll be talking about now is strictly for the left hand side. Now, the first thing I'll do is recall that co tangent of negative is equal to cosine of negative X divided by sine of negative X. And I can also recall that cosine of negative X is equal to cosine of X. And that sign of negative X is equal to a negative sign of X. Now, using all of that information, I can rewrite our equation and we're left with open parentheses, cosine of X divided by negative sign of X closed parentheses multiplied by cosine of X. And now, if I distribute that I'll be left with negative cosine squared X divided by sign of X. And I'll stop here because maybe I can get the right hand side of our equation to this point. And that's exactly what we'll try now. So now we'll talk about the right hand side of our equation. So the first thing I'll do is recall once again, in this case, we'll recall that cos cans of X is equal to one divided by sine of X and I can rewrite with that and we rewrite it as sin of X minus one divided by sin of X. And now to turn these into one fraction, I gotta find a common denominator to be able to combine them. And that common denominator will be sin of X. So I multiply the sin of X on the left um by a fraction of sin of X divided by sin of X and will be left with in the numerator of a new of our new fraction will be sine squared X minus one divided by the denominator which is still sin of X. Now, I can recall that sine squared X is equal to one minus cosine squared X. This is thanks to our python, I, that's these and I can rewrite. And now in the numerator, we'll have one minus cosine squared X minus one divided by sine of X. And now the one, we have a one minus one which is zero. So we're going to be left with negative cosine squared X divided by sine of X which matches the left hand side of our equation. So we have negative cosine squared X divided by sine X is equal to negative cosine squared X divided by sin of X, which means the answer is a true. So I hope that was helpful. And until next time.

In Exercises 1–60, verify each identity. tan (-x) cos x = -sin x

Key Concepts

Trigonometric Identities

Tangent Function and Its Properties

Negative Angle Identities

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