ProofHard4 marks
Prove that (1 + cot θ - cosec θ)(1 + tan θ + sec θ) = 2
Topic: Trigonometric identities
Previous-year question practice database
Answer
Exam answer
Left-hand side = (1+cotθ−cosecθ)(1+tanθ+secθ)
= [(sinθ+cosθ−1)/sinθ] [(cosθ+sinθ+1)/cosθ]
= [(sinθ+cosθ)²−1]/(sinθcosθ)
= [sin²θ+cos²θ+2sinθcosθ−1]/(sinθcosθ)
= 2.
Hence Left-hand side = Right-hand side.
Explanation
Convert all functions to sine and cosine; the numerator becomes a difference of squares. Use sin²θ+cos²θ=1.
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