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O-Level Trigonometric functions, identities and equations: worked solution
4 marks. Full working, one step per line.
Question
Show that (cos A − sin A)(1 + sin A cos A) divided by sin³A equals cot³A − 1.
Worked answer
RHS: cot³ A - 1 = cos³ A/sin³ A - 1 = (cos³ A - sin³ A)/sin³ A. Factor the difference of cubes: cos³ A - sin³ A = (cos A - sin A)(cos² A + cos A sin A + sin² A). Since cos² A + sin² A = 1, this = (cos A - sin A)(1 + sin A cos A). Therefore (cos A - sin A)(1 + sin A cos A)/sin³ A = (cos³ A - sin³ A)/sin³ A = cot³ A - 1. (shown)
Practise this topic
This question is part of O-Level Trigonometric functions, identities and equations, in O-Level Additional Maths (A-Maths).
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