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Question
Show that:
`sqrt((1-cos"A")/(1+cos"A"))=cos"ecA - cotA"`
Solution
LHS = `sqrt((1-cos"A")/(1+cos"A"))`
⇒ LHS = `sqrt((1-cos"A")/(1+cos"A")xx (1-cos"A")/(1-cos"A"))`
⇒ LHS = `sqrt(((1-cos"A")^2)/(1-cos^2"A"))`
=`sqrt(((1-cos"A")^2)/sin^2"A")` ....(sin2A =1- cos2A)
⇒ LHS = `sqrt(((1-cos"A")/sin"A")^2)`
⇒ LHS = `(1-cos"A")/sin"A" = 1/sin"A" + cos"A"/sin"A"`
⇒ LHS = `cos"ecA"-cot"A"` = RHS
Hence Proved
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Proof: L.H.S. = (sec θ – cos θ) (cot θ + tan θ)
= `(1/square - cos θ) (square/square + square/square)` ......`[∵ sec θ = 1/square, cot θ = square/square and tan θ = square/square]`
= `((1 - square)/square) ((square + square)/(square square))`
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= `square/(square square)`
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= R.H.S.
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∴ (sec θ – cos θ) (cot θ + tan θ) = tan θ.sec θ
