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प्रश्न
Prove the following identities:
`(cosecA)/(cosecA - 1) + (cosecA)/(cosecA + 1) = 2sec^2A`
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उत्तर
L.H.S. `=(cosec A)/(cosecA - 1) + (cosecA)/(cosecA + 1)`
= `(cosec A (cosec A + 1) + cosec A (cosec A - 1))/((cosec A - 1) (cosec A + 1))`
= `(cosec^2 A+cosec A + cosec^2 A-cosec A)/((cosec A)^2 - (1)^2)`
= `(2 cosec^2 A)/(cosec^2 A - 1)`
= `(2 cosec^2 A)/(cot^2 A)` ...(∵ cosec2 A – 1 = cot2 A)
= `2(1/cancel(sin^2A))/(cos^2A/cancel(sin^2A))`
= `2/cos^2A`
= 2 sec2 A
= R.H.S.
संबंधित प्रश्न
Show that `sqrt((1+cosA)/(1-cosA)) = cosec A + cot A`
Prove the following trigonometric identities.
`(1 + cos θ + sin θ)/(1 + cos θ - sin θ) = (1 + sin θ)/cos θ`
Prove the following trigonometric identity.
`(sin theta - cos theta + 1)/(sin theta + cos theta - 1) = 1/(sec theta - tan theta)`
Prove the following identities:
`(sec A - 1)/(sec A + 1) = (1 - cos A)/(1 + cos A)`
Prove the following identities:
(1 – tan A)2 + (1 + tan A)2 = 2 sec2A
Find the value of sin ` 48° sec 42° + cos 48° cosec 42°`
If \[\sin \theta = \frac{4}{5}\] what is the value of cotθ + cosecθ?
Prove the following identity :
`(1 - tanA)^2 + (1 + tanA)^2 = 2sec^2A`
Prove that
sin2A . tan A + cos2A . cot A + 2 sin A . cos A = tan A + cot A
Complete the following activity to prove:
cotθ + tanθ = cosecθ × secθ
Activity: L.H.S. = cotθ + tanθ
= `cosθ/sinθ + square/cosθ`
= `(square + sin^2theta)/(sinθ xx cosθ)`
= `1/(sinθ xx cosθ)` ....... ∵ `square`
= `1/sinθ xx 1/cosθ`
= `square xx secθ`
∴ L.H.S. = R.H.S.
