Infinitude Conditions for Sophie Germain Primes

This paper explores Sophie Germain primes of the form 2p+1 where, p is a prime. By extension, the paper also explores other properties of prime numbers. I derive a trigonometric product identity that isolates the condition for a prime p to form a Sophie Germain prime pair, i.e. such that 2p+1 is also prime. The analysis shows that the classical tangent–sine product expansion, when modified by the divisor-sum function, sigma(n), reproduces a constant equality only under this primality constraint. By taking the logarithm of the product, the result reduces to a convergent series of finite trigonometric sums involving cosecant powers and Bernoulli-number polynomials. This establishes an analytic equivalence between Sophie Germain primality and harmonic cancellation within a sine lattice determined by the divisor structure of 2p+1.