Sinθsecθ+1+sinθsecθ-1 = 2 cot θ हे सिद्ध करा. - Mathematics 2 - Geometry [गणित २ - भूमिती]

`sintheta/(sectheta+ 1) +sintheta/(sectheta - 1)` = 2 cot θ हे सिद्ध करा.

Sum

डावी बाजू = `sintheta/(sectheta+ 1) +sintheta/(sectheta - 1)`

= `sintheta/(1/costheta + 1) + sintheta/(1/costheta - 1`

= `sintheta/((1 + costheta)/costheta) + sintheta/((1 - costheta)/(costheta))`

= `(sintheta costheta)/(1 + costheta) + (sintheta costheta)/(1 - costheta)`

= `sin theta costheta (1 /(1 + costheta) + 1/(1 - costheta))`

= `sintheta costheta [(1 - costheta + 1 + costheta)/((1 + costheta)(1 - costheta))]`

= `sintheta costheta (2/(1 - cos^2theta))` ......[∵ (a + b)(a – b) = a² – b²]

= `sintheta costheta xx 2/(sin^2theta)` .....`[(because sin^2theta + cos^2theta = 1),(therefore 1 - cos^2theta = sin^2theta)]`

= `2 xx (costheta)/(sintheta)`

= 2cot θ

= उजवी बाजू

∴ `sintheta/(sectheta+ 1) +sintheta/(sectheta - 1)` = 2 cot θ

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