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Question

Prove the following identities.
cos(x+y)=coshxcoshy+sinhxsinhy

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Solution

We know that;
coshx=ex+ex2

sinhx=exex2

coshxcoshy=12(ex+ex)×12(ey+ey)=14(ex+y+exy+e(xy)+e(x+y))

sinhxsinhy=12(exex)×12(eyey)=14(ex+yexye(xy)+e(x+y))

Subtracting gives:
coshxcoshy+sinhxsinhy=2×14(e(x+y)+e(x+y))=12(e(x+y)+e(x+y))=cosh(x+y)
Hence;
cosh(x+y)=coshxcoshy+sinhxsinhy


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