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Question
Prove the following identity :
( 1 + cotθ - cosecθ) ( 1 + tanθ + secθ)
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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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