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Verify: sinh(x + y) = sinh xcoshy + cosh x sinh yHint: Use the definition sinh and cosh...

Question

Verify: sinh(x + y) = sinh xcoshy + cosh x sinh yHint: Use the definition sinh and cosh

Verify: sinh(x + y) = sinh xcoshy + cosh x sinh y Hint: Use the definition sinh and cosh



Answers

Verify the following identities. $$\sinh (x+y)=\sinh x \cosh y+\cosh x \sinh y$$

Yeah, so and again. Problem we have to find were toe prove that the left hand side this request to write inside. So we will start from the right hand side the light inside this course hyperbolic X called hyperbolic y plus sine hyperbolic, exciting hyperbolically. We just have to use the definition off the course hyperbolic and sine hyperbolic in the in the exponential form. So we'll convert this into exponential form cause hyperbolic X. Will it cost to e to the power X plus to the minus X by to cause hyperbolic y will become equal to eat part of I minus he to the power. So it plus you to the par minus way divided by two, the sine hyperbolic X will become an exponential form. It will be to the products minus here to depart minus x by two and China, the polic y will be e to the power of I minus here to about minus y by toe. When we multiply this and we regroup these terms, we will get, uh A to the power X plus y plus eight to Dubai minor self X plus y. There are that by toe. So from the definition off course, hyperbolic. We can write it. That's cause hyperbolic X plus y, which is the left hand side, so it is pulled. That left hand side is request toe, light and side.

Okay. We first started out with either the excuse y minus even negative X plus y looked closely at where the parentheses are. Gives us you did the X minus you the negative acts over too. Times eat of the Y, plus even negative. Why over too? Plus, each of the AKs puts even negative acts over too times either the y minus e to the negative. Why over to So as you can see, the night gives and the positives are alternating when we're doing this copulation. And this friendly gives us sign a jukebox co sign a chew of Why was co sign h of X sign each of why.

All right. We want to identify or whether verify the identity. Cinch or hyperbolic sine of X plus Y is equal to a hyperbolic sine X times hyperbolic coastline. Y plus the hyperbolic sine X times a hyperbolic sine Y. This question is challenging our knowledge of the transcendental functions known as a hyperbolic functions. Remember the hydraulic functions are synched, X equals x minus X. Over to coach ecstasy. The X plus X over to chance essential to coach and coach squared minus X squared equals one. We're going to rewrite the right hand side to match the left hand side to verify his identity. So since ex coach Y plus coach X and Y is one half times E. To the X minus negative extends either Y plus Y plus either the X plus either negative X times either the y minus either negative Y, distributing these multiplication and then simplifying gifts. One half of the X plus y minus negative x minus y. However, this is simply the form of a hyperbolic sine of X plus Y, which means the two sides of our equation are equal and our identity is verified.

We want to prove me some formula or cinch by changing the expression to an exponential. Now, for this, I think it might be easier for us to start on the right hand side and then work towards the left hand side. So the right hand side, Yes, you too. Well, let's just go ahead and plug everything in so cinch of X. So we have the definitions of each upto. So it be e to the X minus e to the negative X over. Too cautious. Why is going to be easy to the why plus e to the negative? Why over too And then it would just be reversed on the other side. So b e to the X plus e to the negative x over too times e to the y, minus e to the negative light over too well, notice. Each will have a denominator of four. So let's just go ahead and factor out 1/4 and then we can oil these individually so boiling the numerator is here. This would give us E to the X plus. Why must be to the X minus y and minus a to the why minus X and then plus Oh, the lost will actually be negative minus e to the negative x minus y. And in our other one, when we foil this side out, we would get lost so e to the x plus y and minus hey to the X minus y then plus G to the live minus sechs and then minus e to the negative x minus y. Now let's see if we can combined any like terms. So we have he to the ex wa e to the X plus y there and then we have the e to the X plus y here so those can combined to give us to e to the X plus y, uh then we have e to the X minus y here. And then we have another one here, but noticed those are opposite signs. So those just cancel out with each other. We have a E to the Y, minus sechs here. Then we have a e to the wind my sex years of those to cancel out and then lastly, me how an e to the negative x minus y here as well as here. So that would become minus two e to the negative X minus. What? So the 1/4 of the twos can simplify down too? Just perhaps we have e to the x plus y minus e. And I'm gonna factor a negative out of this power here to write it like this. And it be over too. And now we have the difference. Oh, the two exponential sze, which falls into the case of cinch. So this is now going to be equal to cinch about X plus one. So we proved what they wanted us to use a probable proof box and smile, please.


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