Solution #18

We can rewrite Σ_n=1⁴⁴cos n as Σ_n=1⁴⁴cos (45-x). (This just reverses the direction of the addends.) We then use the cosine angle sum identity to rewrite cos (45-x) as (cos 45)(cos x) + (sin 45)(sin x). Since cos 45 = sin 45 = 1/ √ 2  , we can rewrite this as 1/ √ 2 (cos x + sin x).

This gives us:

Σ_n=1⁴⁴cos n = 1/ √ 2 Σ_n=1⁴⁴(cos n + sin n) = 1/ √ 2(Σ_n=1⁴⁴cos n) + 1/ √ 2(Σ_n=1⁴⁴sin n).

Thus:

(1 - 1/ √ 2)Σ_n=1⁴⁴cos n = 1/ √ 2(Σ_n=1⁴⁴sin n)

(Σ_n=1⁴⁴cos n)/(Σ_n=1⁴⁴sin n) = 1/ √ 2 /(1 - 1/ √ 2) = 1 + √ 2