J's answer to mirabelle's Secondary 3 A Maths Singapore question.
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1/(1- sinθ) + secθ + tanθ
= 1/(1 - sinθ) × (1 + sinθ)/(1 + sinθ) + 1/cosθ + sinθ/cosθ
= (1 + sinθ)/[(1 - sinθ)(1 + sinθ)] + 1/cosθ × cosθ/cosθ + sinθ/cosθ × cosθ/cosθ
= (1 + sinθ)/(1² - sin²θ) + cosθ/cos²θ + sinθcosθ/cos²θ
= (1 + sinθ)/cos²θ + cosθ/cos²θ + sinθcosθ/cos²θ
= (1 + sinθ + cosθ + sinθcosθ)/cos²θ
= (1 + sinθ + cosθ(1 + sinθ))/cos²θ
= (1 + sinθ)(1 + cosθ)/cos²θ
(Proved)
= 1/(1 - sinθ) × (1 + sinθ)/(1 + sinθ) + 1/cosθ + sinθ/cosθ
= (1 + sinθ)/[(1 - sinθ)(1 + sinθ)] + 1/cosθ × cosθ/cosθ + sinθ/cosθ × cosθ/cosθ
= (1 + sinθ)/(1² - sin²θ) + cosθ/cos²θ + sinθcosθ/cos²θ
= (1 + sinθ)/cos²θ + cosθ/cos²θ + sinθcosθ/cos²θ
= (1 + sinθ + cosθ + sinθcosθ)/cos²θ
= (1 + sinθ + cosθ(1 + sinθ))/cos²θ
= (1 + sinθ)(1 + cosθ)/cos²θ
(Proved)
Date Posted:
3 years ago
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