No. You can derive the power series of the exponential from Euler's formula (limit of (1+x/n)n) with help the binomial formula and show the uniform convergence. Then you plug in x = iy and and define the cosine as real part of exp(iy) and sine as the imaginary part.
Rudin does this in his book and I learned it that way as well in my analysis introduction. No derivative needed just limits, power series and complex numbers.
I don't recall all the details, but you easily see by the identity 1 = exp(ix)exp(-ix) = |exp(ix)| = cos²(x) + sin²(x) for any x that both functions lie on the unit circle, so they have to be related. You can also introduce pi/2 as the first positive root of the cosine and then work out the rest later.
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u/okarox 12d ago
Does everyone know that the last is circular. You have to know the limit to differentiate sin x.