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Question
If cosecθ = `1(9)/(20)`, show that `(1 - sinθ + cosθ)/(1 + sinθ + cosθ) = (3)/(7)`
Solution
cosecθ = `1(9)/(20) = (29)/(20)`
sinθ = `(1)/(cosecθ) = (20)/(29) = "Perpendicular"/"Hypotenuse"`
Base
= `sqrt(("Hypotenuse")^2 - ("Perpendicular")^2`
= `sqrt((29)^2 - (20)^2`
= `sqrt(841 - 400)`
= `sqrt(441)`
= 21
cosθ = `"Base"/"Hypotenuse" = (21)/(29)`
To show: `(1 - sinθ + cosθ)/(1 + sinθ + cosθ) = (3)/(7)`
`(1 - sinθ + cosθ)/(1 + sinθ + cosθ)`
= `(1 - 20/29 + 21/29)/(1 + 20/29 + 21/29)`
= `(29 - 20 + 21)/(29 + 20 + 21)`
= `(30)/(70)`
= `(3)/(7)`.
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