ProofHard3 marks
Prove that: sin A / (1 + cot A) + cos A / (1 + tan A) = sin A - cos A
Topic: Trigonometric identities
Previous-year question practice database
Answer
Exam answer
Left-hand side = sinA/(1+cotA) − cosA/(1+tanA)
= sin²A/(sinA+cosA) − cos²A/(sinA+cosA)
= (sin²A−cos²A)/(sinA+cosA)
= [(sinA−cosA)(sinA+cosA)]/(sinA+cosA)
= sinA−cosA = Right-hand side.
Hence proved.
Explanation
Convert cotA and tanA to cosine/sine forms, combine over the common denominator, and factor the difference of squares.
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