Show that (cos 2θ)/(sin θ cos θ) ≡ cot θ - tan θ.

Mark scheme

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

1. (a) M1: Uses the double angle formula for cos 2θ = cos² θ - sin² θ (or 2 cos² θ - 1 or 1 - 2 sin² θ).
2. (a) M1: Divides all terms by sin θ cos θ. E.g., (cos² θ)/(sin θ cos θ) - (sin² θ)/(sin θ cos θ).
3. (a) A1: Simplifies to (cos θ)/(sin θ) - (sin θ)/(cos θ) ≡ cot θ - tan θ. (Must show intermediate step to reach cot θ - tan θ)
4. (b) M1: Uses the result from part (a) to write (cos 2x)/(sin x cos x) = 3 or 2(cos 2x)/(2 sin x cos x) = 3 implies (2 cos 2x)/(sin 2x) = 3.
5. (b) M1: Rearranges into a form that can be solved, e.g., cot 2x = 3/2 or tan 2x = 2/3.
6. (b) A1: Finds the principal value for 2x. tan 2x = 2/3 implies 2x = arctan(2/3) ≈ 33.69...circ. Finds at least one correct value for 2x in the range. (Range for 2x is 0 < 2x < 360circ). So 2x = 33.69...circ, 33.69...+180circ = 213.69...circ.
7. (b) A1: Correct values for x to one decimal place: x = 16.8circ and x = 106.8circ. (Must round correctly)