Evaluate:

Concept: For any angle x, the half-angle identities 2 + 2cos x = 4cos2(x/2) and 2 − 2cos x = 4sin2(x/2) hold; taking principal square roots turns √(2 + 2cos x) into 2|cos(x/2)| and √(2 − 2cos x) into 2|sin(x/2)|, and these drop to 2cos(x/2) and 2sin(x/2) whenever x/2 is acute — as it is at every step below. Chaining these, the product identity 2 sinθ · 2 cosθ = 2 sin2θ collapses two adjacent half-angle factors into a single sine of the doubled angle — repeating this shortens a whole chain of nested square roots down to one clean sine value.

Application:

1. Read the printed expression as four factors multiplied together: √(2 + √3), √(2 + √(2 + √3)), √(2 + √(2 + √(2 + √3))) and √(2 − √(2 + √(2 + √3))).

2. Since √3 = 2cos30°, the innermost piece gives 2 + √3 = 2 + 2cos30° = 4cos215°, so √(2 + √3) = 2cos15°.

3. Feed this outward one layer at a time with the same identity: √(2 + 2cos15°) = 2cos7.5°, then √(2 + 2cos7.5°) = 2cos3.75°.

4. The last, differently-signed factor uses the sine version instead: √(2 − 2cos7.5°) = 2sin3.75°.

5. Multiply the four factors and collapse them from the inside out using 2sinθ·2cosθ = 2sin2θ: (2cos3.75°)(2sin3.75°) = 2sin7.5°.

6. (2cos7.5°)(2sin7.5°) = 2sin15°.

7. (2cos15°)(2sin15°) = 2sin30° = 2 × ½ = 1.

Cross-check: Evaluating each nested factor as a decimal confirms this — √(2+√3) ≈ 1.9319, √(2+√(2+√3)) ≈ 1.9829, √(2+√(2+√(2+√3))) ≈ 1.9957, and √(2−√(2+√(2+√3))) ≈ 0.1308 — multiplying all four gives ≈ 1.000, matching the telescoped result.

So the product of all four factors equals 1.