sin(30°) = 1/2 and cos(60°) = 1/2, thus the sum is 1/2 + 1/2 = 1.

Let θ = cos^(-1)(1/2). Then cos(θ) = 1/2, which corresponds to θ = π/3. Therefore, sin(θ) = sin(π/3) = √3/2.

sin(tan^(-1)(1)) = 1/√2, using the triangle with opposite and adjacent sides equal.

Using the right triangle definition, sin(tan^(-1)(x)) = opposite/hypotenuse = x/√(1+x^2).

Since π/3 is in the range of sin^(-1), sin^(-1)(sin(π/3)) = π/3.

Since sin(π/4) = 1/√2, sin^(-1)(1/√2) = π/4.

Since sin^2(θ) = 1 - cos^2(θ), we have sin^(-1)(√(1 - cos^2(θ))) = sin^(-1)(sin(θ)) = θ.

sin^(-1)(√3/2) = π/3 and cos^(-1)(1/2) = π/3. Therefore, the sum is π/3 + π/3 = 2π/3.

sin^(-1)(√3/2) = π/3 and sin^(-1)(1/2) = π/6. Therefore, π/3 + π/6 = π/2.

sin^(-1)(√3/2) = π/3.

The coefficient of x^2 is C(4,2) * (3)^2 * (-4)^2 = 6 * 9 * 16 = 864.

f'(x) = 4x^3 - 12x^2 + 12x. Evaluating at x = 1 gives f'(1) = 4 - 12 + 12 = 4.

The determinant is \( 1*4 - 2*3 = 4 - 6 = -2 \).

The determinant is calculated as 1(1*0 - 4*6) - 2(0*0 - 4*5) + 3(0*6 - 1*5) = 1(0 - 24) - 2(0 - 20) + 3(0 - 5) = -24 + 40 - 15 = 1.

Using the determinant formula, we calculate it to be 1.

Using the determinant formula, we calculate it to be -10.

Using the determinant formula, we calculate it to be -20.

Substituting x = 1 gives the determinant | 1  1  2 |  | 3  1  4 |  | 5  6  1 | = 6.

The integral evaluates to [x - (1/3)x^3] from 0 to 1 = 1 - 1/3 = 2/3.

The integral ∫(3x^2)dx from 0 to 1 = [x^3] from 0 to 1 = 1.

The integral evaluates to [(1/3)x^3 + x^2] from 0 to 1 = (1/3 + 1) - 0 = 4/3.

Subtracting 5 from both sides gives 3x = 15, thus x = 15/3 = 5.

Using the quadratic formula, z = [-4 ± √(16 - 32)]/2 = -2 ± 2i.

Taking square root, z = ±√(-16) = ±4i.

z^2 = (1 + i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i.

Setting the determinant to zero gives \( k = 6 \).

The determinant of the identity matrix is always 1.

By definition, \( \sin(\sin^{-1}(x)) = x \). Therefore, \( \sin(\sin^{-1}(\frac{1}{2})) = \frac{1}{2} \).

The integral evaluates to [x - x^3/3] from 0 to 1 = (1 - 1/3) = 2/3.

The integral evaluates to [e^x] from 0 to 1 = e - 1.