Question

If sin (x + iy) sin (e + id) = 1, show that
(i) tanh? y cosh? o = cos2 0;
(ii) tanh? o cosh” y = cos” x.​

Answer 1

Answer:

Step-by-step explanation:

I begin with

sin(x+iy)=ex+iy−e−x−iy2i=exeiy−e−xe−iy2i

=exeiy−e−xeiy+−e−xeiy−e−xe−iy2i

=eiy(ex−e−x)−e−xeiy−e−xe−iy2i

=eiysin(x)+−e−xeiy−e−xe−iy2i

At this point I could have −e−xicos(iy) in the right term but I want to make cos(x) appear, so I don't know how to continue.

For all x,y∈R the following holds:

sin(x+iy)=ei(x+iy)−e−i(x+iy)2i=e−y+ix−ey−ix2i=e−y(cos(x)+isin(x))−ey(cos(x)−isin(x))2i=(e−y−ey)cos(x)+i(e−y+ey)sin(x)2i=(e−y−ey)cos(x)2i+i(e−y+ey)sin(x)2i=isinh(y)cos(x)+cosh(y)sin(x