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प्रश्न
sin2A . tan A + cos2A . cot A + 2 sin A . cos A = tan A + cot A हे सिद्ध करा.
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उत्तर
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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संबंधित प्रश्न
sec θ(1 - sin θ) (sec θ + tan θ) = 1
sec2θ + cosec2θ = sec2θ × cosec2θ
खालील प्रश्नासाठी उत्तराचा योग्य पर्याय निवडा.
sin2θ + sin2(90 – θ) = ?
cot2θ × sec2θ = cot2θ + 1 हे सिद्ध करा.
tan2θ – sin2θ = tan2θ × sin2θ हे सिद्ध करण्यासाठी खालील कृती पूर्ण करा.
कृती: डावी बाजू = `square`
= `square (1 - (sin^2theta)/(tan^2theta))`
= `tan^2theta (1 - square/((sin^2theta)/(cos^2theta)))`
= `tan^2theta (1 - (sin^2theta)/1 xx (cos^2theta)/square)`
= `tan^2theta (1 - square)`
= `tan^2theta xx square` .....[1 – cos2θ = sin2θ]
= उजवी बाजू
`(tan(90 - theta) + cot(90 - theta))/("cosec" theta)` = sec θ हे सिद्ध करा.
`sintheta/(sectheta+ 1) +sintheta/(sectheta - 1)` = 2 cot θ हे सिद्ध करा.
sin4A – cos4A = 1 – 2cos2A हे सिद्ध करा.
`(1 + sec "A")/"sec A" = (sin^2"A")/(1 - cos"A")` हे सिद्ध करा.
जर cos A + cos2A = 1, तर sin2A + sin4A = ?
