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Question
Prove the following identity :
( 1 + cotθ - cosecθ) ( 1 + tanθ + secθ)
Solution
(1 + cotθ - cosecθ) ( 1 + tanθ + secθ)
= `(1 + sinθ/cosθ + 1/cosθ)(1 + cosθ/sinθ - 1/sinθ)`
= `((cosθ + sinθ + 1)/cosθ)((sinθ + cosθ - 1)/sinθ)`
= `((sinθ + cosθ)^2 - (1)^2)/(sinθcosθ)`
= `(sin^2θ + cos^2θ + 2sinθ cosθ - 1)/(sinθcosθ)`
= `(1 + 2sinθ cosθ - 1)/(sinθcosθ)`
= `(2sinθ cosθ)/(sinθ cosθ) = 2`
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Complete the following activity to prove:
cotθ + tanθ = cosecθ × secθ
Activity: L.H.S. = cotθ + tanθ
= `cosθ/sinθ + square/cosθ`
= `(square + sin^2theta)/(sinθ xx cosθ)`
= `1/(sinθ xx cosθ)` ....... ∵ `square`
= `1/sinθ xx 1/cosθ`
= `square xx secθ`
∴ L.H.S. = R.H.S.
