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Prove That: Sqrt(( Secθ - 1)/(Secθ + 1)) + Sqrt((Secθ + 1)/(Secθ - 1)) = 2cosecθ - Mathematics

Sum

Prove that:
`sqrt(( secθ - 1)/(secθ + 1)) + sqrt((secθ + 1)/(secθ - 1)) = 2cosecθ`

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Solution

LHS = `sqrt(( secθ - 1)/(secθ + 1)) + sqrt((secθ + 1)/(secθ - 1))` 

= `(sqrt( secθ - 1) sqrt( secθ - 1) + sqrt( secθ + 1)sqrt( secθ + 1))/(sqrt(secθ - 1)sqrt(secθ + 1))`

= `((sqrt( secθ - 1))^2 + (sqrt( secθ + 1))^2)/(sqrt(secθ - 1)sqrt(secθ + 1))`

= `(secθ - 1 + secθ + 1)/(sqrt(sec^2 - 1))`

= `(2secθ)/sqrt(tan^2θ)`

= `(2secθ)/(tanθ)`

= `(2 1/cosθ)/(sinθ/cosθ)`

= `(2 1/sinθ)`

= 2 cosecθ.

Concept: Trigonometric Identities
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