Use de Moivre's theorem to express cot 7θ in terms of cot θ. Use the equation cot 7θ = 0 to show that the roots of the equation x⁶ − 21x⁴ + 35x² − 7 = 0 are cot²((kπ)/14) for k = 1, 3, 5, 9, 11, 13, and deduce that cot²(π/14) cot²(3π/14) cot²(5π/14) = 7.

Use de Moivre's theorem to show that tan 5θ = (5t - 10t³ + t⁵) / (1 - 10t² + 5t⁴), where t = tan θ. Deduce that the r...

By considering ∑_{r=1}^{n} z²ʳ⁻¹, where z = cos θ + i sin θ, show that, if sin θ ≠ 0, ∑_{r=1}^{n} sin (2r − 1)θ = sin...

Let z = cos θ + i sin θ.

The equation x³ + 2x² + x + 7 = 0 has roots α, β, γ.

The matrix M is defined by M = [[2,m,1],[0,m,7],[0,0,1]] where m ≠ 0, 1, 2.

The matrix A is defined by A = [[1,5,1],[1,-2,-2],[2,3,θ]]