Show that sin³x (cosec x / cotx) can be written as sin²x tanx.

Mark scheme

Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.

1. [M1] Convincing correct statement from which the answer can be easily determined e.g. sin³x (1/sinx) / (cosx/sinx) oe
2. [M1] or sin²x * (sinx / (sinx cotx)) oe
3. [M1] or sin³x (1/sinx * tanx) oe
4. [M1] or sin³x (cosec x tanx) oe
5. [M1] or sin³x (cosec x / (cosx / sinx)) oe
6. [M1] or sin³x ( (1/sinx) / (cosx / sinx) ) oe
7. [A1] Correct completion to given answer: sin²x tanx
8. [M1] Factorises: tanx(cos²x - 1/2) = 0 oe or tanx(1 - sin²x - 1/2) = 0 oe or tanx(1/2 - sin²x) = 0 oe or sinx cos²x - (1/2)sinx = 0 oe or sinx(cos²x - 1/2) = 0 oe or sinx(1 - sin²x - 1/2) = 0 oe or tanx(1-tan²x) = 0 oe
9. [A1] [tanx = 0 or sinx = 0] [x =] 0
10. [M1] nfww
11. [M1] cosx = [±]1/√2 oe or sinx = [±]1/√2 oe or tanx = [±]1
12. [A2] [x = ] π/4 or ±0.785 or ± 0.7853 to ± 0.7854, 3π/4 or ±2.36 or ± 2.356 to ± 2.3562 with no extras in range; nfww
13. [A1] for any two out of four correct, ignoring extras