cos(θ) = 1/2 at θ = 30° and 150°.

Using the identity sin^2(θ) + cos^2(θ) = 1, we have sin^2(θ) = 1 - (1/2)^2 = 1 - 1/4 = 3/4. Thus, sin(θ) = ±√3/2.

cos(30°) = 1/2, hence θ = 30°.

cos(60°) = 1/2, so θ = 60°

From cos^(-1)(x) = θ, we have cos(θ) = x. Therefore, sin(θ) = √(1 - cos^2(θ)) = √(1 - x^2).

Cotangent is 1 when the angle A is 45°, as cot A = cos A/sin A = 1 when both are equal.

cot A = cos A/sin A = 3/4 implies sin A = 4/5 using the identity sin²A + cos²A = 1.

cot D = 3/4 implies tan D = 4/3. Therefore, sin D = 4/5 (using the identity sin²D + cos²D = 1).

cot θ = 1 implies tan θ = 1, which occurs at θ = 45°.

Using the identity cot θ = cos θ / sin θ, we find sin θ = 4/5.

Using the identity cot θ = cos θ / sin θ, we can find sin θ = 12/13 using the Pythagorean theorem: sin θ = 12/√(5^2 + 12^2) = 12/13.

Using the identity cot(x) = cos(x)/sin(x), we can find sin(x) = 12/13 using the Pythagorean theorem.

Since D = E and E > F, it follows that D > F.

The trace of a matrix is the sum of its diagonal elements. Trace(D) = 2 + 2 = 4.

The inverse of D is given by (1/det(D)) * adj(D). Here, det(D) = (4*3) - (2*1) = 10, and adj(D) = [[3, -2], [-1, 4]]. Thus, D^(-1) = (1/10) * [[3, -2], [-1, 4]].

The inverse of D is given by (1/det(D)) * adj(D). Here, det(D) = (4*3) - (2*1) = 10. The adjugate is [[3, -2], [-1, 4]]. Thus, D^(-1) = (1/10) * [[3, -2], [-1, 4]].

The number of ways to choose 3 elements from a set of 6 is given by the combination formula C(n, r) = n! / (r!(n-r)!). Here, C(6, 3) = 20.

The total number of subsets of D is 2^4 = 16. Proper subsets are all subsets except the set itself, so there are 16 - 1 = 15 proper subsets.

The total number of subsets is 2^4 = 16. Proper subsets exclude the set itself, so 16 - 1 = 15.

A subset must contain elements from the original set. {4} is not a subset of D.

The proper subsets of D = {1, 2} are {∅}, {1}, and {2}. So, there are 3 subsets, but excluding D itself gives us 2 proper subsets.

The power set of D = {1, 2} is {∅, {1}, {2}, {1, 2}} which contains 4 subsets.

A subset can only contain elements from the original set. {3} is not a subset of D.

A subset can only contain elements from the original set. {1} is not a subset of D.

The number of subsets of a set with n elements is 2^n. Here, n = 2, so 2^2 = 4.

The subsets containing 'a' are {a}, {a, b}, so there are 2 subsets.

Since D is opposite E, F sitting to the left of D means F is also opposite to E. Therefore, the person sitting to the right of E is F.

The arrangement can be visualized as A-B-C-E-D-F. Therefore, A is sitting to the left of E.

F is the son of G, making G the grandfather of D.

D is the mother of E, and since E is the sister of F, D is also the mother of F.