#Product formula of cosh(x/2^k)

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wispy zenith
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Hello.
I want to know how to prove the formula for the product of cosh(x/2^k), which is 2^-n × sinh(x)/sinh(x/2^n). I can notice the similarity with the regular cosine formula but we proved that one with complex numbers and I don't know how to use those in here. As I've just studied these functions, I don't know every formula and the ones I know don't help. Please help and thanks!

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plush dragon
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@wispy zenith

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sure

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cosh(2kx​)=2−k⋅sin(2kx​)sinh(x)​

wispy zenith
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I don't understand the transition from the 5th to 6th picture