Prove the identity tan x + 1/tan x = 1/(sin x cos x)

Mark scheme

Marking points from the official mark scheme (May/June 2012). The mark or mark type is shown in brackets.

1. [M1] LHS = sin x / cos x + cos x / sin x; Use of tan = sin/cos twice.
2. [M1] = (sin² x + cos² x) / (sin x cos x) = 1 / (sin x cos x); Use of s² + c² = 1 appropriately – everything correct.
3. [M1] Uses (i) to substitute 1/(sin x cos x) = tan x + 1/tan x, leading to 2(tan x + 1/tan x) = 1 + 3 tan x; Uses part (i) to obtain eqn in tanx only.
4. [DM1] Rearranges to form a quadratic equation tan² x + tan x - 2 = 0; Correct soln of quadratic eqn.
5. [B1 A1] Solves quadratic to get tan x = 1 or tan x = -2; co. Must have correct quadratic co.
6. [A1] Solutions x = 45° or x = 116.6°; (One mark for each correct solution, or both for A1 for both and no others in range.)