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Question
If cosec θ = `(29)/(20)`, find the value of: cosec θ - `(1)/("cot" θ)`
Solution
Consider ΔABC, where ∠A = 90°
⇒ cosec θ = `"Hypotenuse"/"Perpendicular" = "BC"/"AB" = (29)/(20)`
By Pythagoras theorem,
BC2 = AB2 + AC2
⇒ AC2 = BC2 - AB2
= 292 - 202
= 841 - 400
= 441
⇒ AC = 21
Now,
sec θ = `"Hypotenuse"/"Base" = "BC"/"AC" = (29)/(21)`
tan θ = `"Perpendicular"/"Base" = "AB"/"AC" = (20)/(21)`
⇒ cot θ = `(1)/"tan θ " = (21)/(20)`
`"cosec" θ - (1)/"cot θ"`
= `(29)/(20) - (1)/(21/20)`
= `(29)/(20) - (20)/(21)`
= `(609 - 400)/(420)`
= `(209)/(420)`.
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