Sin2A . tan A + cos2A . cot A + 2 sin A . cos A = tan A + cot A हे सिद्ध करा. - Mathematics 2 - Geometry [गणित २ - भूमिती]

sin²A . tan A + cos²A . cot A + 2 sin A . cos A = tan A + cot A हे सिद्ध करा.

Sum

डावी बाजू = sin²A . tan A + cos²A . cot A + 2 sin A . cos A

= `sin^2"A"* (sin "A")/(cos "A") + cos^2"A"* (cos"A")/(sin"A") + 2sin"A" *cos"A"`

= `(sin^3"A")/"cosA" + (cos^3"A")/"sinA" + 2sin"A"*cos"A"`

= `(sin^4"A" + cos^4"A" + 2sin^2"A"cos^2"A")/(sin"A"cos"A")`

= `(sin^2"A" + cos^2"A")^2/(sin"A"cos"A")` .....[∵ a² + b² + 2ab = (a + b)²]

= `1^2/(sin"A"cos"A")` ......[∵ sin²A + cos²A = 1]

= `1/(sin"A"cos"A")`

= `(sin^2"A"+ cos^2"A")/(sin"A"cos"A")` ......[∵ 1 = sin²A + cos²A]

= `(sin^2"A")/(sin"A"cos"A") + (cos^2"A")/(sin"A"cos"A")`

= `"sinA"/"cosA" + "cosA"/"sinA"`

= tan A + cot A

= उजवी बाजू

∴ sin²A . tan A + cos²A . cot A + 2 sin A . cos A = tan A + cot A

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