O lvl A Maths: Trigonometric Identity

a) Prove that (tan A + cot A)(sin A + cos A) ≡ 1/cos A + 1/sin A

b) Prove that sec A ≡ sin 2A / sin A - cos 2A / cos A

c) Prove that cot² A - cos² A ≡ cot² A cos² A

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Answer:

a) LHS = (tan A + cot A)(sin A + cos A)
= tan A sin A + tan A cos A + cot A sin A + cot A cos A
= (sin A / cos A) sin A + (sin A / cos A) cos A + (cos A / sin A) sin A + (cos A / sin A) cos A
= (sin² A / cos A) + sin A + cos A + (cos² A / sin A)
= (sin² A / cos A) + (sin² A / sin A) + (cos² A / cos A) + (cos² A / sin A)
= (sin² A + cos² A) / cos A + (sin² A + cos² A) / sin A
= 1 / cos A + 1 / sin A
= RHS (proved)

b) RHS = sin 2A / sin A - cos 2A / cos A
= (2 sin A cos A) / sin A - (2 cos² A - 1) / cos A
= 2 cos A- 2 cos A + 1 / cos A
= sec A
= LHS (proved)

c) LHS = cot² A - cos² A
= cos² A / sin² A - (cos² A sin² A) / sin² A
= { cos² A ( 1 - sin² A) } / sin² A
= cos² A (cos² A / sin² A)
= cos² A cot² A
= RHS (proved)

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