If sin A + cos A = 1 / (2√2), then find the value of sin^4 A + cos^4 A.

Question

Accepted Answer

Given: sin A + cos A = 1 / (2√2)

Step 1: Square both sides:
(sin A + cos A)^2 = (1 / (2√2))^2
sin^2 A + 2 sin A cos A + cos^2 A = 1 / 8

Step 2: Use identity sin^2 A + cos^2 A = 1:
1 + 2 sin A cos A = 1 / 8

Step 3: Solve for sin A cos A:
2 sin A cos A = 1/8 - 1 = -7/8
sin A cos A = -7/16

Step 4: Find sin^4 A + cos^4 A using the identity:
sin^4 A + cos^4 A = (sin^2 A + cos^2 A)^2 - 2 sin^2 A cos^2 A
= 1^2 - 2 (sin A cos A)^2
= 1 - 2 (-7/16)^2
= 1 - 2 (49/256)
= 1 - 98/256
= (256/256) - (98/256) = 158/256

Step 5: Simplify the fraction:
158/256 = 79/128

Therefore, sin^4 A + cos^4 A = 79/128.

Answer

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