Using the Pythagorean identity, cos C = √(1 - sin² C) = √(1 - 0.25) = √0.75 = √3/2.

Using the Pythagorean identity, cos C = √(1 - sin²C) = √(1 - 0.8²) = √(1 - 0.64) = √(0.36) = 0.6.

Using the Pythagorean identity, cos x = √(1 - sin²x) = √(1 - 0.6²) = √(1 - 0.36) = √(0.64) = 0.8.

Using the Pythagorean identity, cos θ = √(1 - sin²θ) = √(1 - 0.8²) = √(1 - 0.64) = √(0.36) = 0.6.

Using the identity, sin²A = 1 - cos²A = 1 - (0.6)² = 1 - 0.36 = 0.64.

If cos A = 0, then sin²A = 1 - cos²A = 1 - 0 = 1.

Since tan θ = sin θ / cos θ, and tan θ = 1/√3, we can use the 30-60-90 triangle where sin θ = 1/2.

Height = distance * tan(30) = 100 * (1/√3) ≈ 57.74 m, which rounds to 50 m.

Height = shadow * tan(angle) = 10 * tan(30) = 10 * (1/√3) ≈ 10 m.

Height = Shadow * tan(30) = 10√3 * (1/√3) = 10 m.

x² + y² = (x + y)² - 2xy = 10² - 2*21 = 100 - 42 = 58.

x² - y² = 4² - 5² = 16 - 25 = -9.

(x - 1)² = (5 - 1)² = 4² = 16.

(2x - 3)² = 4x² - 12x + 9 by the square of a binomial.

(2x - 3y)² = 4x² - 12xy + 9y² by applying the square of a binomial.

(x + 2)(x - 2) = x² - 2² = x² - 4 by the difference of squares.

(2x + 3)(2x - 3) = 4x² - 9 by the difference of squares.

(a - b)(a + b) = a² - b², which is the difference of squares.

(x + 1)² - (x - 1)² = [(x² + 2x + 1) - (x² - 2x + 1)] = 4x.

(x + y + z)² = x² + y² + z² + 2(xy + xz + yz) is the correct expansion.

(x + y)(x - y) = x² - y², which is the difference of squares.

(2x + 3)(2x - 3) = 4x² - 9.

(2x + 3)² = 4x² + 12x + 9 by expanding the square of a binomial.

(3x + 4)(3x - 4) = (3x)² - 4² = 9x² - 16.

(a + b)² - (a² + b²) = 2ab.

(a + b)² = a² + 2ab + b² by the expansion of the square of a binomial.

(x + 1)((x + 2) + (x - 2)) = (x + 1)(2x) = 2x² + 2x.

(x + 1)(x - 1) = x² - 1, which is the difference of squares.

(x + 2)(x - 2) = x² - 4 by the difference of squares identity.

(x + 2)² = x² + 4x + 4; thus, (x + 2)² - (x² + 4) = 4x.