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प्रश्न
Prove the following trigonometric identities.
`sqrt((1 - cos theta)/(1 + cos theta)) = cosec theta - cot theta`
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उत्तर
We know that, `sin^2 theta + cos^2 theta = 1`
Multiplying numerator and denominator under the square root by `1 - cos theta)` we have
`sqrt((1 - cos theta)/(1 + cos theta)) = sqrt(((1 - cos theta)(1 - cos theta))/((1 + cos theta)(1 - cos theta)))`
`= sqrt((1 - cos theta)^2/(1 - cos^2 theta))`
`= sqrt((1 - cos theta)^2/sin^2 theta`
`= (1 - cos theta)/sin theta`
`= 1/sin theta - cos theta/sin theta`
`= cosec theta - cot theta`
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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
Proved that `(1 + secA)/secA = (sin^2A)/(1 - cos A)`.
