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प्रश्न
Prove that `sqrt((1 + cos theta)/(1 - cos theta)) + sqrt((1 - cos theta)/(1 + cos theta)) = 2 cosec theta`
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उत्तर
`sqrt((1 + cos theta)/(1 - cos theta)) + sqrt((1 - cos theta)/(1 + cos theta))`
`= sqrt((1 + cos theta)/(1 - cos theta) xx (1 + cos theta)/(1 + cos theta)) + sqrt((1 -cos theta)/(1 + cos theta) xx (1 - cos theta)/(1 - cos theta))`
`= sqrt((1 + cos theta)^2/(1 - cos^2 theta)) + sqrt((1 - cos theta)^2/(1 - cos^2 theta))`
`= sqrt((1 + cos theta)^2/(sin^2 theta)) + sqrt((1 -cos theta)^2/sin^2 theta)`
`= (1 + cos theta)/sin theta + (1 - cos theta)/sin theta`
`= 2/sin theta = 2cosec theta`
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Activity: L.H.S. = cotθ + tanθ
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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
