Using the double angle formula sin(2x) = 2sin x cos x. Since sin x = 1/2, cos x = √(1 - (1/2)^2) = √(3/4) = √3/2. Thus, sin(2x) = 2 * (1/2) * (√3/2) = √3/2.

Using the double angle formula sin(2θ) = 2sin θ cos θ. First, find cos θ using sin^2 θ + cos^2 θ = 1. cos θ = sqrt(1 - (1/3)^2) = sqrt(8/9) = 2sqrt(2)/3. Thus, sin(2θ) = 2 * (1/3) * (2sqrt(2)/3) = 4sqrt(2)/9.

Using the double angle formula, sin(2θ) = 2sin(θ)cos(θ) = 2(1/√2)(1/√2) = 1.

Using the double angle formula, sin(2θ) = 2sin θcos θ.

Using the double angle formula, sin(2θ) = 2sin(θ)cos(θ).

sin(30 degrees) = 1/2 = 0.5.

sin(30°) = 1/2 and cos(60°) = 1/2. Therefore, sin(30°) + cos(60°) = 1/2 + 1/2 = 1.

sin(30°) = 1/2.

sin(45°) = cos(45°) = √2/2. Therefore, sin(45°) + cos(45°) = √2/2 + √2/2 = √2.

sin(45°) = 1/√2

Using the co-function identity, sin(90° - A) = cos A.

Using the co-function identity, sin(90° - x) = cos(x).

sin(90°) = 1.

sin(tan^(-1)(1)) = √2/2

sin(tan^(-1)(x)) = x/√(1+x^2)

sin(tan^(-1)(√3)) = √3/2

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

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

sin^(-1)(1/2) = π/6, since sin(π/6) = 1/2.

sin^(-1)(sin(π/4)) = π/4, as π/4 is in the range of sin^(-1).

By the Pythagorean identity, sin^2(x) + cos^2(x) = 1 for all x.

By the Pythagorean identity, sin^2(θ) + cos^2(θ) = 1.

According to the Pythagorean identity, sin² θ + cos² θ = 1 for any angle θ.

tan(0°) = 0

tan(30°) = sin(30°)/cos(30°) = (1/2)/(√3/2) = 1/√3.

Using the formula tan(A + B) = (tan A + tan B) / (1 - tan A tan B), we have tan(45° + x) = (1 + tan x) / (1 - 1*tan x) = (1 + tan x).

tan(45°) = 1.

tan(60°) = √3.

tan(90°) is undefined.

tan^(-1)(1) = π/4, thus tan^(-1)(1) + tan^(-1)(1) = π/4 + π/4 = π/2.