(i) Prove that `tan θ + cot θ ≡ sec θ cosec θ`. (ii) Solve, for `0 ≤ x < 360circ`, the equation `2 tan x - 3 cot x - 1 = 0`.

Mark scheme

Marking points from the official mark scheme. The mark or mark type is shown in brackets.

1. M1 for expressing `tan θ` and `cot θ` in terms of `sin θ` and `cos θ`. e.g. `sin θ / cos θ + cos θ / sin θ`.
2. A1 for combining terms, e.g. `(sin² θ + cos² θ) / (sin θ cos θ)`.
3. A1 for `1 / (sin θ cos θ) = sec θ cosec θ`.
4. M1 for using `cot x = 1/tan x` to form an equation in `tan x`. e.g. `2 tan x - 3/tan x - 1 = 0`.
5. M1 for forming a 3 term quadratic in `tan x`. e.g. `2 tan^2 x - tan x - 3 = 0`.
6. A1 for `(2 tan x - 3)(tan x + 1) = 0`.
7. A1 for `tan x = 3/2` and `tan x = -1`.
8. B1 for one correct angle `x=45^\circ` or `x=56.3^\circ` (accept `56.31^\circ`).
9. M1 for using `180 - α` or `360 - α` to find second solution for either `tan x = 3/2` or `tan x = -1`.
10. A1 for all four correct angles `45circ, 225circ, 56.3circ, 236.3circ` (allow `56.3circ` to `56.4circ`, `236.3circ` to `236.4circ`). Must be `0 ≤ x < 360circ`.