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Proving simple identities sheet
3-mark answer frame

Prove that cos A/(1 − tan A) + sin A/(1 − cot A) = sin A + cos A.

How to lay out the answer. Follow the frame, then write it in your own words.

  1. 1 · Write what is given and what is to be found To prove: LHS = cos A/(1 − tan A) + sin A/(1 − cot A) equals RHS = sin A + cos A.
  2. 2 · Write the formula or theorem tan A = sin A/cos A, cot A = cos A/sin A, a² − b² = (a − b)(a + b).
  3. 3 · Change to sin and cos LHS = cos²A/(cos A − sin A) + sin²A/(sin A − cos A).
  4. 4 · Make the denominators the same = cos²A/(cos A − sin A) − sin²A/(cos A − sin A) = (cos²A − sin²A)/(cos A − sin A).
  5. 5 · Factorise and cancel = (cos A − sin A)(cos A + sin A)/(cos A − sin A) = cos A + sin A.
  6. 6 · Check and box the answer = sin A + cos A = RHS. Write 'Hence proved'.

Scoring tip: Notice that sin A − cos A = −(cos A − sin A). Turning the second fraction round this way gives a common denominator in one line.