Question

Find sin^4(a) + cos^4(a), if cos(a) + sin(a) = 1/3

Answer 1

Answer:

49/81

Step-by-step explanation:

[cos(a) + sin(a)]^2 = (1/3)^2

(cos(a))^2 + 2sin(a)cos(a) + (sin(a))^2 = 1/9

(sin(a))^2 + (cos(a))^2 = 1

1 + 2sin(a)cos(a) = 1/9

2sin(a)cos(a) = -8/9

sin(a)cos(a) = -4/9

[cos(a) + sin(a)]^4 = (1/3)^4 = 1/81

(cos(a))^4 + 4sin(a)×(cos(a))^3 + 6×(sin(a))^2×(cos(a))^2 + 4(sin(a))^3×cos(a) + (sin(a))^4 = 1/81

(cos(a))^4 + (sin(a))^4 + 4sin(a)cos(a)((cos(a))^2 + (sin(a))^2) + 6(sin(a)cos(a))^2 = 1/81

cos(a))^4 + (sin(a))^4 + 4sin(a)cos(a)(1) + 6(sin(a)cos(a))^2 = 1/81

(cos(a))^4 + (sin(a))^4 + 4(-4/9) +6((-4/9)^2) = 1/81

(cos(a))^4 + (sin(a))^4 - 16/9 + 6(16/81) = 1/81

(cos(a))^4 + (sin(a))^4 = 1/81 + 16/9 - 6(16/81)

(cos(a))^4 + (sin(a))^4 = 49/81