Question

For any angle θ, the expression `(2 cos 8θ + 1)/(2 cos θ + 1)` is equal to ______.

Options

• (2 cos θ + 1) (2 cos 2θ + 1) (2 cos 4θ + 1)

• (cos θ – 1) (cos 2θ – 1) (cos 4θ – 1)

• (2 cos θ – 1) (2 cos 2θ – 1) (2 cos 4θ – 1)

• (2 cos θ + 1) (2 cos 2θ + 1) (2 cos 4θ + 1)

Answer 1

For any angle θ, the expression `(2 cos 8θ + 1)/(2 cos θ + 1)` is equal to (2 cos θ – 1) (2 cos 2θ – 1) (2 cos 4θ – 1).

Explanation:

We have, `(2 cos 8θ + 1)/(2 cos θ + 1)`

= `(2(2 cos^2 4θ - 1) + 1)/((2 cos θ + 1))`

= `((2 cos 4θ - 1)(2 cos 4θ + 1))/((2 cos θ + 1))`

= `((2 cos 4θ - 1)(2 cos 2θ - 1)(2 cos 2θ + 1))/((2 cos θ + 1))`

= `([(2 cos 4θ - 1)(2 cos 2θ - 1)(2 cos θ - 1)(2 cos θ + 1)])/((2 cos θ + 1))`

= (2 cos 4θ – 1)(2 cos 2θ – 1)(2 cos θ – 1)