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प्रश्न
Prove that `sin A/(sec A + tan A - 1) + cos A/(cosec A + cot A - 1) = 1`.
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उत्तर
LHS = `(sec A)/(sec A + tan A - 1) + cos A/(cosec A + cot A - 1)`
= `(sin A)/(1/cos A + sin A/cos A - 1) + cos A/(1/sin A + cos A/sin A - 1)`
= `(sin A/(1 + sin A - cos A))/cos A + ((cos A)/(1 + cos A - sin A))/(sin A)`
= `(sin A.cos A)/(1 + sin A - cos A) + (sin A. cos A)/(1 + cos A - sin A)`
= `(sin A. cos A( 1 + cos A - sin A + 1 + sin A - cos A))/([ 1 + (sin A - cos A)][1 - (sin A - cos A)])`
= `(2sin A. cos A)/((1)^2 - (sin A - cos A)^2)`
= `(2sin A. cos A)/(1 - (sin^2 A + cos^2 A - 2 sin A.cos A))`
= `(2 sin A. cos A)/(1 - 1 + 2 sin A. cos A)`
= `2/2 = 1`
= RHS
Hence proved.
संबंधित प्रश्न
Prove the following identities, where the angles involved are acute angles for which the expressions are defined:
`(cosec θ – cot θ)^2 = (1-cos theta)/(1 + cos theta)`
Prove the following trigonometric identities.
(1 + tan2θ) (1 − sinθ) (1 + sinθ) = 1
Prove the following trigonometric identities.
`(1 + cos A)/sin A = sin A/(1 - cos A)`
Prove that:
`sqrt(sec^2A + cosec^2A) = tanA + cotA`
`(1+ cos theta)(1- costheta )(1+cos^2 theta)=1`
Show that none of the following is an identity:
(i) `cos^2theta + cos theta =1`
If cosec2 θ (1 + cos θ) (1 − cos θ) = λ, then find the value of λ.
Prove the following identity :
`(1 + cosA)/(1 - cosA) = (cosecA + cotA)^2`
Prove that sec2θ + cosec2θ = sec2θ × cosec2θ.
Prove the following:
`sintheta/(1 + cos theta) + (1 + cos theta)/sintheta` = 2cosecθ
