Let θ = tan^(-1)(1). Then, cos(θ) = 1/√(1 + tan^2(θ)) = 1/√(1 + 1) = 1/√2.

Using the triangle with opposite = 3 and adjacent = 4, hypotenuse = 5. Thus, cos(tan^(-1)(3/4)) = 4/5.

Using the right triangle definition, cos(tan^(-1)(5/12)) = adjacent/hypotenuse = 12/13.

sin(cos^(-1)(1/2)) = √(1 - (1/2)^2) = √(3/4) = √3/2.

Using the right triangle definition, sin(tan^(-1)(3/4)) = opposite/hypotenuse = 3/5.

Using the identity, sin(tan^(-1)(x)) = x/√(1+x^2).

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

Since 5π/6 is in the range of sin^(-1), sin^(-1)(sin(5π/6)) = 5π/6.

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

sin^(-1)(√3/2) + cos^(-1)(1/2) = π/2

Let θ = sin^(-1)(3/5). Then sin(θ) = 3/5 and using the Pythagorean theorem, cos(θ) = 4/5. Therefore, tan(θ) = sin(θ)/cos(θ) = (3/5)/(4/5) = 3/4.

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

tan^(-1)(1) = π/4 and tan^(-1)(√3) = π/3. Therefore, π/4 + π/3 = π/2.

sin^(-1)(1) = π/2 and cos^(-1)(0) = π/2. Therefore, π/2 + π/2 = π.

tan^(-1)(1) = π/4, so the expression becomes π/4 + π/4 + 0 = π/2.

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

tan^(-1)(1) = π/4 and tan^(-1)(√3) = π/3. Therefore, π/4 + π/3 = 7π/12.

sin^(-1)(0) = 0 and cos^(-1)(0) = π/2, thus the sum is 0 + π/2 = π/2.

sin^(-1)(1) = π/2 and cos^(-1)(0) = π/2. Therefore, π/2 + π/2 = π.

cos(tan^(-1)(1)) = 1/√2

cos(tan^(-1)(3)) = 3/√10

cos^(-1)(-1/2) = 2π/3, since cos(2π/3) = -1/2.

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

sin^(-1)(√3/2) = π/3.