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secondary 3 | A Maths
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Date Posted: 3 years ago
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(2sinx + sin 2x) / (2sinx - sin2x)
= (2sinx + 2sinxcosx) / (2sinx - 2sinxcosx)
(Recall that sin2A = 2sinAcosA)
= (1 + cosx) / (1 - cosx)
(Divide both numerator and denominator by 2sinx)
= [(1 + cosx)(1 + cosx)] / [(1 - cosx)(1 + cos x)]
(Rationalising the denominator)
= (1² + 2cosx + cos²x) / (1² - cos²x)
(Numerator : (a + b)² = a² + 2ab + b²
Denominator : (a - b)(a + b) = a² - b² )
= (1 + 2cosx + cos²x) / (sin²x)
(Properties : 1 - cos²x = sin²x and 1² = 1)
= 1/sin²x + 2cosx/sin²x + cos²x/sin²x
= (1/sinx)² + 2(1/sinx)(cosx/sinx) + (cosx/sinx)²
= cosec²x + 2cosecx cotx + cot²x
(Proved)
= (2sinx + 2sinxcosx) / (2sinx - 2sinxcosx)
(Recall that sin2A = 2sinAcosA)
= (1 + cosx) / (1 - cosx)
(Divide both numerator and denominator by 2sinx)
= [(1 + cosx)(1 + cosx)] / [(1 - cosx)(1 + cos x)]
(Rationalising the denominator)
= (1² + 2cosx + cos²x) / (1² - cos²x)
(Numerator : (a + b)² = a² + 2ab + b²
Denominator : (a - b)(a + b) = a² - b² )
= (1 + 2cosx + cos²x) / (sin²x)
(Properties : 1 - cos²x = sin²x and 1² = 1)
= 1/sin²x + 2cosx/sin²x + cos²x/sin²x
= (1/sinx)² + 2(1/sinx)(cosx/sinx) + (cosx/sinx)²
= cosec²x + 2cosecx cotx + cot²x
(Proved)
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