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

1. 1 · Write what is given and what is to be found To prove: LHS = RHS. Start from the LHS, because it is the more complicated side.
2. 2 · Write the identities you will use tan θ = sin θ/cos θ, cot θ = cos θ/sin θ, and a² − b² = (a + b)(a − b).
3. 3 · Simplify the first term cos θ/(1 − sin θ/cos θ) = cos θ/((cos θ − sin θ)/cos θ) = cos²θ/(cos θ − sin θ)
4. 4 · Simplify the second term sin θ/(1 − cos θ/sin θ) = sin²θ/(sin θ − cos θ) = −sin²θ/(cos θ − sin θ)
5. 5 · Add the two terms LHS = (cos²θ − sin²θ)/(cos θ − sin θ) = (cos θ + sin θ)(cos θ − sin θ)/(cos θ − sin θ)
6. 6 · Check and box the answer LHS = cos θ + sin θ = RHS. Hence proved.

Scoring tip: Make both denominators the same, (cos θ − sin θ), by taking out a minus sign. That one step unlocks the proof.