prove-that-e-z-e-z-2-2-cos-2-y-sinh-2-x-when-z-x-iy-

Question Number 88263 by M±th+et£s last updated on 09/Apr/20

p r o v e t h a t

∣ \frac{e^{z} - e^{- z}}{2} ∣^{2} + {c o s}^{2} y = {s i n h}^{2} x w h e n z = x + i y

Commented by M±th+et£s last updated on 09/Apr/20

p l e a s e h e l p

Answered by TANMAY PANACEA. last updated on 09/Apr/20

e^{x + i y} = e^{x} . e^{i y} = (c o s h x + s i n h x) (c o s y + i s i n y)

= c o s h x c o s y + i c o s h x s i n y + s i n h x c o s y + i s i n h x s i n y

= c o s y (c o s h x + s i n h x) + i s i n y (c o s h x + s i n h x)

e^{- (x + i y)} = e^{- x} . e^{- i y} = (c o s h x - s i n h x) (c o s y - i s i n y)

= c o s h x c o s y - i c o s h x s i n y - s i n h x c o s y + i s i n h x s i n y

= c o s y (c o s h x - s i n h x) - i s i n y (c o s h x - s i n h x)

n o w

\frac{e^{z} - e^{- z}}{2} = \frac{1}{2} [c o s y (2 s i n h x) + i s i n y (2 c o s h x)]

∣ \frac{e^{z} - e^{- z}}{2} ∣^{2} = {(c o s y s i n h x)}^{2} + {(s i n y c o s h x)}^{2}

{c o s}^{2} y {(s i n h x)}^{2} + {s i n}^{2} y {(c o s h x)}^{2} + {c o s}^{2} y

= {c o s}^{2} y [1 + {(s i n h x)}^{2}] + {s i n}^{2} y {(c o s h x)}^{2}

= {c o s}^{2} y [{c o s}^{2} h x - {s i n}^{2} h x + {s i n}^{2} h x] + {s i n}^{2} {y c o s}^{2} h x

= ({c o s}^{2} y + {s i n}^{2} y) {c o s}^{2} h x

= {c o s}^{2} h x

p l s c h e c k

Commented by M±th+et£s last updated on 09/Apr/20

y e s s i r i t s t y p o g o d b l e s s y o u

Commented by TANMAY PANACEA. last updated on 09/Apr/20

m o s t w e l c o m e s i r