Final answer:
The expression (sin(7π/24) + cos(7π/24))^2 can be simplified using trigonometric identities to 1 + sin(7π/12). By breaking down 7π/12 into sum of known angles and applying sum identities of sine and cosine, we obtain the exact simplified value.
Step-by-step explanation:
To simplify the expression (sin(7π/24) + cos(7π/24))^2, we can use the trigonometric identity for the sum of squares of sine and cosine that states: sin^2(x) + 2 sin(x) cos(x) + cos^2(x) = 1 + 2 sin(x) cos(x). Applying this to our expression yields:
- sin^2(7π/24) + 2sin(7π/24)cos(7π/24) + cos^2(7π/24).
- Since sin^2(x) + cos^2(x) equals 1, we can simplify this to 1 + 2sin(7π/24)cos(7π/24).
- Using the double-angle identity for cosine, which is cos(2x) = 1 - 2sin^2(x), we can rewrite 2sin(x)cos(x) as sin(2x). So the expression becomes 1 + sin(7π/12).
To find the exact value, we can rewrite 7π/12 as the sum or difference of angles for which we know the sine values:
- sin(7π/12) = sin(3π/4 + π/6), which can be further simplified using sum of angles identity for sine: sin(A + B) = sin(A)cos(B) + cos(A)sin(B).
- This leads to sin(3π/4)cos(π/6) + cos(3π/4)sin(π/6).
- After evaluating, we'll have the exact simplified value for our original expression.