Alternative to solve this?

  • Thread starter MrWarlock616
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In summary, by expanding the given expression and equating real and imaginary parts, we can obtain the equations ##\cos(\alpha)=\sin(x)\cosh(y)## and ##\sin(\alpha)=\cos(x)\sinh(y)##. By using the identity ##\cos^2(\alpha)+\sin^2(\alpha)=1##, we can prove the expression ##\cosh(2y)-\cos(2x)=2##. Alternatively, we can use the identity ##cosh(a)=cos(i a)## to simplify the expression and prove it in a different way.
  • #1
MrWarlock616
160
3
I need to prove that if ##\sin(x+iy)=cos(\alpha)+i\sin(\alpha)## , then ##\cosh(2y)-\cos(2x)=2##

expanding sin(x+iy) and equating real and imaginary parts, we get:
##\cos(\alpha)=\sin(x)\cosh(y)## -- (1)
and ## \sin(\alpha)=\cos(x)\sinh(y) ## -- (2)

if we use ##\cos^2(\alpha)+\sin^2(\alpha)=1 ## and put the values from (1) and (2) , we can get the required expression. My question is that, is there an alternative to do this? Can I prove it by directly using the values of cosh2y and cos2x [from (1) and (2)]? I tried but it seems impossible. Why is it so?
 
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  • #2
use
cosh(a)=cos(i a)
then
cos(2a)-cos(2b)=-2sin(a-b)sin(a+b)
and so forth
 
  • #3
lurflurf said:
use
cosh(a)=cos(i a)
then
cos(2a)-cos(2b)=-2sin(a-b)sin(a+b)
and so forth

ok..what is a and b?
 
  • #4
Whatever you want (those are identities), in this case perhaps
cosh(2y)=cos(2i y)
cos(2iy)-cos(2x)=-2sin(iy-x)sin(iy+x)
 
  • #5
lurflurf said:
Whatever you want (those are identities), in this case perhaps
cosh(2y)=cos(2i y)
cos(2iy)-cos(2x)=-2sin(iy-x)sin(iy+x)

Ok that is brilliant. thanks!
 

1. What is an alternative solution?

An alternative solution is a different approach or method to solving a problem or achieving a goal. It is an option that is different from the traditional or commonly used solution.

2. Why would we need an alternative solution?

There are many reasons why an alternative solution may be necessary, such as the current solution being ineffective, too expensive, or not feasible. It may also be needed to address new challenges or changes in the environment.

3. How do we determine the best alternative solution?

The best alternative solution should be evaluated based on its effectiveness, efficiency, feasibility, and impact. It should also be compared to other potential solutions to determine which one is the most suitable.

4. Are alternative solutions always better than traditional solutions?

Not necessarily. While alternative solutions may have benefits, they may also have drawbacks and limitations. It is important to carefully evaluate and consider all factors before deciding on the best solution.

5. Can alternative solutions be applied to all problems?

No, not all problems can have alternative solutions. Some problems may have specific constraints or limitations that do not allow for alternative solutions to be effective. It is important to assess the problem thoroughly before considering alternative solutions.

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