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Question
Prove that: `sqrt((1 - cos θ)/(1 + cos θ)) = cosec θ - cot θ`.
Solution
LHS = `sqrt((1 - cos θ)/(1 + cos θ) xx (1 - cos θ)/(1 - cos θ))`
= `sqrt((1 - cos θ)^2/(1 - cos^2θ))`
= `(1 - cos θ)/(sqrt(1 - cos^2θ))`
= `(1 - cos θ)/(sqrt(sin^2θ))`
= `(1 - cos θ)/(sin θ)`
= `(1)/(sin θ) - (cos θ)/(sin θ)`
= cosec θ - cot θ
= RHS
Hence proved.
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Show that, cotθ + tanθ = cosecθ × secθ
Solution :
L.H.S. = cotθ + tanθ
= `cosθ/sinθ + sinθ/cosθ`
= `(square + square)/(sinθ xx cosθ)`
= `1/(sinθ xx cosθ)` ............... `square`
= `1/sinθ xx 1/square`
= cosecθ × secθ
L.H.S. = R.H.S
∴ cotθ + tanθ = cosecθ × secθ
