As θ takes every real value, the least value of $2^{si{n}^2\theta }+2^{co{s}^2\theta }$ is:

The two exponents always add to 1, so the product of the two powers is fixed at $2^{si{n}^2\theta }\times 2^{co{s}^2\theta }=2^1=2$. For a pair of positive numbers with a fixed product the sum is least when they are equal, and AM–GM makes that precise: the sum is at least 2√2 ≈ 2.828, reached when $si$$n^2$θ = $co$$s^2$θ = 1/2, $i$.e. θ = 45°, where the value is √2 + √2. The value 3 is what the expression gives at θ = 0° (namely 1 + 2) — that is the largest value, not the least; 2 would need both powers to be 1, which the fixed sum of exponents forbids; and a spot check at θ = 30° gives 2.871, safely above 2.828.