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Prove that (𝑚 + 1)𝑃𝑚+1(𝑦) − 𝑦𝑃𝑚(𝑦) − (2𝑚)𝑦𝑃𝑚(𝑦) ...

Question

Prove that (𝑚 + 1)𝑃𝑚+1(𝑦) − 𝑦𝑃𝑚(𝑦) − (2𝑚)𝑦𝑃𝑚(𝑦) + 𝑚𝑃𝑚−1(𝑦) =0.

Prove that (𝑚 + 1)𝑃𝑚+1(𝑦) − 𝑦𝑃𝑚(𝑦) − (2𝑚)𝑦𝑃𝑚(𝑦) + 𝑚𝑃𝑚−1(𝑦) = 0.



Answers

Prove the following identities. $$\sin ^{-1} y+\sin ^{-1}(-y)=0$$

Foods we have. Scientific right? San data equals equals. Why? Right where the domain data lies between zero by over two. Then when I take him for a sign of either side of the equation, this becomes that, right? And then eso you haven't ever sign of eggs off. Why? As they're right here, right again, if I have Ah, son is a, uh is an odd function, Right? So this one is actually gonna be negative. Sign it is how you do art functions, right? So, uh, if I have this one, then this one is actually, you know, we said that Cynthia is why, right? So negative science data is that they're gonna be negative. Why, Right, So you have this one. So we will now add the two here, right? So if negative scientist A is negative way, then when I take the inverse sign off on either side of the equation, right, take the increasing off inverse side of this equation of both sides. Then I'm gonna have negative theater here, right? Representing number. Sign off. Negative. Why here, Like that, Right. So now I want to add it to right? So if I want to add it to here. Uh, if I want to add in verse, sign off. Why? Plus members sign off into a sign of negative white here. Right? I'm adding this one and that one. Right. Okay, so we know that in verse. Sign off. Why is actually data here? Right. So I'm gonna put it here as data and an ember sign of negative way is negative data here. Right. So I'm gonna put that on this negative data. Then what am I gonna get? Theorem in? Its data is actually zero, right. So in re sign off, White was younger. Sign of negative. Why is actually zero?

Okay. The first thing to notice here today is that we can write co sign princes expose pi over three as co sign X plus heil to minus pi over six. We know what we can further simplify this to be sign of pi over sex minus axe, which we can write as negative in terms of negative sign. Therefore, we know that co sign princes exports by over three with signed parentheses. X over Poseidon purposes X minus pi over six equals zero. And the reason why is because we know that we can rewrite this part over here as negative sign prophecies. X minus pi over sex. So now if I've just simplified the left hand side is negative. Sign X minus pi over six. And I bring this part over here. Down on the right hand side. Negative sign expense by over six. Plus positive sign X. My spy over six is just gonna be zero. So now we have zero on the left hand side and we bring down zero on the right hand side. So it's We've just proved it. Andy

Here we need to prove that a dirt Crosby. This should be close to do this. We need to prove as we know that Crosby is equals two. Give me scientific every sentence dependent, so he'll be signed two days of activities perpendicular to both A and B. And this implies that it is also perpendicular to. So we can we can read E. Lord eight rows B. This should be equals to zero, since mm and equals B are perpendicular hands. We can, late Crosby equals to zero. So we have proved student when.

Okay, so for this problem we have to prove this equation. Okay, so let's look at the left side. Okay, let's expand this left side. Okay, so we have this it goes to a C plus B C. Times I plus 80 times I plus be the Hi squared OK? Which he goes to A C. Okay. Plus these two terms we can come by right. So we have factor out I that we have B C plus A. D. And for this term okay, I squared equals two minus one. So we have minus B. D. Okay, so let's group this and this. So we have a c minus B. D plus. Okay, B C plus or you can see A D plus B C. Okay, A D plus B C. I. Okay, so you can see this is exactly the same as the right side of the equation. Okay, so we we prove that this equation stands


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