Answer to Question #122814 in Trigonometry for Ojugbele Daniel

Question

Expand cos6theta and sin6theta in terms of costheta and sintheta

Answer 1

Let x = cos⁶(θ) + sin⁶(θ)

x = (cos²(θ))³ + (sin²(θ))³

i.e., x = (cos²(θ) + sin²(θ)) × ( (cos²(θ))² + (sin²(θ))² - cos²(θ)sin²(θ))

x = (1) × ( (cos²(θ))² + (sin²(θ))² - cos²(θ)sin²(θ) + 2cos²(θ)sin²(θ) - 2cos²(θ)sin²(θ) )

x = ( (cos²(θ))² + (sin²(θ))² + 2cos²(θ)sin²(θ) ) - 3cos²(θ)sin²(θ)

i.e., x = (cos²(θ) + sin²(θ))² - 3cos²(θ)sin²(θ)

$\therefore$ x = 1 - 3cos²(θ)sin²(θ)