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Problem 34 Hard Difficulty

Use the methods of Example 6 to prove the identity
$$ 2\sin^{-1} x = \cos^{-1} (1 - 2x^2) x \geqslant 0 $$

Answer

$$
2 \sin ^{-1} x=\cos ^{-1}\left(1-2 x^{2}\right)
$$

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Video Transcript

Okay, so we're now being asked to use the methods of example sex to prove there's identity right here to sign inverse of X equals co signers of one minus two. X squared is greater than equal to zero for I mean for all ex created unequal issue. Yeah. Ah, Thought what you could start off with doing this is still it affects equal to sign in verse of X minus co sign in verse one of one minus two x squared on DH. Now, the best thing to do is to take the derivative We're going to take a derivative. I have private act, So do it of signing verse. X is one over square of one minus X squared to the constant weaken, right, two over the square root of one minus x squared. And then the next one is a little bit more complicated. We have to apply to change rules. So we take the derivative the inside, which will give us negative for X. So write negative for X times. Then we're going. We can apply. These change will again. So take the derivative cosign inverse, which is negative. One of the square of one minus X Well, but instead of X, we're going right. We're going to put in one minus two Ex Claire, one of the negative one over the square root of one minor, one minus two x squared, squared. And then what we're gonna do is we're gonna simplify. Going to do a little comes in provocation, so they're still have two over square of one minus x squared, and then this is going to become minus four eggs over the square root, and then we're going to expand this. We're going to spend the bottom. We're going to spend one minus two x square. There's going to be one minus one plus Teo, get for for X to the fourth. We're gonna bring this over and then sent to We're going to get four minus for X squared. And this is our we'LL also go the next page for more space. Andi can further simplify this bye and it reminder this is a crime. I'm back, doesn't determinative. We're going to minus four eggs over. I see we have once you're going to cancels and it's just going to be for X for my niche for X squared. They were just going to be left with the square were of negative for X to the fourth plus for X squared. Yeah, push for X squared. And then if you look at this, we can actually factor out a four x squared so we can rewrite this as f back. This is going to be written at minus four. Eggs over this square for X squared times one minus X squared. Received that. That's the same statement. Uh, and then you can actually bring how there, uh, for X squared. You can bring this hour because the squared a forced to discard of X, Where is X? But be careful. When you pull out a r, you pull out like a variable out of a square root, you have to put it in the absolute value cymbal. The Since we're doing what ex positives? Because we're told ex positive numbers, though in any number of greater than zero and after value function ist of function itself. So we're just going to actually cancel, So we're going to cancel, too. This is going to become too, and the exes are going to cancel. So then we are going. Teo, actually now have Primex equal two over the square of one minus X, where minus two over the square root of one mind sex wears well, and we know that this is equal to zero and we're going toe paid. There's that sweet Dirham and these texts and this chapter it's called zero five. You could refer Cletus and fearing five basically states that if f prime vex What if f Crime of X is equal to zero, then after backs and f of X is equal to see. And that makes sense because if you have a river, the secret zero nwe can have the difficult zero If c is the constant. So now, now that we know that effort Max is equal to C, um, we're going to do one more step. We're gonna see apply if we're going to play off of zero to our function, which will give us to sign inverse of zero minus co sign inverse of one minus zero, which is also zero. And this is all we could to see because F of X is equal to C. So So this is going to give you zero. This is also going to give you Teo so then we're left with C is equal to zero. Now we know the actual value of C because he is a constant. I forgot to mention that. So see the mercy of the constant s. So now that we know the actual number, we know that she is constant. So we're gonna move to the next page. We can rewrite the function too. Sign in verse one of eggs minus co. Sign in verse one minus two. Expert is equal to see. But since seeing zero weaken, just write zero. And now all we do is add cause on inverse to the next side. And we have now successfully proven our statement. This is the statement we were looking for. You remember we were asked to prove this identity to sign Inverse of Mexico's goes on your wrist one minus two x where for all excluded and zero. And we can see we have to under