Using de Moivre's theorem, show that tan 5θ = (5 tan θ - 10 tan³ θ + tan⁵ θ) / (1 - 10 tan² θ + 5 tan⁴ θ). Hence show that the equation x² – 10x + 5 = 0 has roots tan²(π/5) and tan²(2π/5). Deduce a quadratic equation, with integer coefficients, having roots sec²(π/5) and sec²(2π/5).

Answer only one of the following two alternatives. EITHER State the fifth roots of unity in the form cos θ + i sin θ,...

Use de Moivre's theorem to show that cos 5θ = cos θ(16 sin⁴ θ – 12 sin² θ + 1). By considering the equation cos 5θ = ...

Let z = cos θ + i sin θ. Show that zⁿ + 1/zⁿ = 2 cos nθ and zⁿ - 1/zⁿ = 2i sin nθ. By considering (z - 1/z)⁴ (z + 1/z...

The variables x and y are such that y = -1 when x = 1 and x² + y² + (dy/dx)³ = 29.

The curve C has polar equation r = a(1 – e⁻ᶿ), where a is a positive constant and 0 ≤ ᶿ < 2π.

At any point (x, y) on the curve C, dx/dt = t(t²+4) and dy/dt = -t√(4-t²), where the parameter t is such that 0 ≤ t ≤ 2.