Using the double angle formula cos(2x) = 1 - 2sin^2 x. Since sin x = 1/2, we have cos(2x) = 1 - 2*(1/2)^2 = 1 - 2*(1/4) = 1 - 1/2 = 1/2.

Using the double angle formula cos(2θ) = 1 - 2sin^2 θ. Here, sin θ = 1/2, so cos(2θ) = 1 - 2(1/2)^2 = 1 - 2(1/4) = 1 - 1/2 = 1/2.

cos(90°) = 0.

cos(π/3) = 1/2.

cos^(-1)(-1) = π, since cos(π) = -1.

cos^(-1)(0) corresponds to the angle where the cosine value is 0, which is π.

cot(cos^(-1)(1/2)) = √3

f(2) = 2^2 - 2(2) + 1 = 4 - 4 + 1 = 1.

i^4 = (i^2)^2 = (-1)^2 = 1.

For equal roots, the discriminant must be zero: k^2 - 4*1*16 = 0, thus k^2 = 64, giving k = -8 or 8. The answer is -4.

The discriminant must be negative: k^2 - 4*1*9 < 0 => k^2 < 36 => |k| < 6, hence k > 6.

Setting 2k + 2 = 0 gives k = 2.

Setting k(0) = 0^2 + 1 gives k = 1.

Setting k(2) = 2^2 gives 2k = 4, thus k = 2.

The minimum value occurs at x = -b/(2a) = -2/(2k). Setting f(-1) = -3 gives k = -2.

For no real roots, the discriminant must be less than zero: k^2 - 4*1*16 < 0 => k^2 < 64 => |k| < 8.

For equal roots, the discriminant must be zero: k^2 - 4*1*16 = 0, thus k^2 = 64, giving k = -8 or k = 8. The answer is -8.

The discriminant must be less than zero: k^2 - 4*1*16 < 0 => k^2 < 64 => k < 8 and k > -8.

For one real root, the discriminant must be zero: k^2 - 4*1*25 = 0, thus k^2 = 100, giving k = -10 or k = 10.

For no real roots, the discriminant must be less than zero: k^2 - 4*1*9 < 0, thus k > 6.

For one real root, the discriminant must be zero: k^2 - 4*1*9 = 0 => k^2 = 36 => k = ±6.

log_10(1000) = 3 and log_10(0.01) = -2, thus 3 - 2 = 1.

log_10(1000) = log_10(10^3) = 3.

log_2(32) = 5 and log_2(4) = 2. Therefore, 5 - 2 = 3.

log_2(32) = 5 and log_2(8) = 3. Therefore, 5 - 3 = 2.

log_3(27) = 3 and log_3(9) = 2. Therefore, 3 - 2 = 1.

log_3(81) = log_3(3^4) = 4.

log_4(64) = log_4(4^3) = 3.

log_5(125) = log_5(5^3) = 3.

log_5(25) = 2 and log_5(5) = 1. Therefore, 2 - 1 = 1.