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प्रश्न
`tan theta /((1 - cot theta )) + cot theta /((1 - tan theta)) = (1+ sec theta cosec theta)`
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उत्तर
LHS= `tan theta/((1-cot theta))+ cot theta/((1-tan theta))`
=`tan theta/((1-cos theta/sin theta)) + cot theta/((1-sin theta/cos theta))`
=`(sin theta tan theta)/((sin theta- cos theta))+(cos theta cot theta)/((cos theta - sin theta))`
=`(sin theta xx (sin theta) / (cos theta) cos theta xx (cos theta) / (sin theta))/((sin theta - cos theta))`
=`((sin ^2 theta cos ^2 theta)/(cos theta sin theta))/((sin theta-cos theta))`
=`( sin ^3 theta - cos ^3 theta)/(cos theta sin theta (sin theta - cos theta))`
=` ((sin theta - cos theta)(sin ^2 theta + sin theta cos theta + cos ^2theta ))/(cos theta sin theta (sin theta- costheta))`
=`(1+ sin theta cos theta)/(cos theta sin theta)`
=`1/(cos theta sin theta)+(sin theta cos theta)/(cos theta sin theta)`
=`1/(cos theta sin theta)+ (sin theta cos theta)/(cos theta sin theta)`
=`sectheta cosec theta +1`
=`1+ sec theta cosec theta`
=RHS
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संबंधित प्रश्न
Prove the following trigonometric identities:
(i) (1 – sin2θ) sec2θ = 1
(ii) cos2θ (1 + tan2θ) = 1
Prove that: `(1 – sinθ + cosθ)^2 = 2(1 + cosθ)(1 – sinθ)`
Prove the following identities, where the angles involved are acute angles for which the expressions are defined:
`(cos A-sinA+1)/(cosA+sinA-1)=cosecA+cotA ` using the identity cosec2 A = 1 cot2 A.
Prove the following trigonometric identities.
`1 + cot^2 theta/(1 + cosec theta) = cosec theta`
Prove the following trigonometric identities.
`(tan A + tan B)/(cot A + cot B) = tan A tan B`
Prove the following identities:
`1/(1 - sinA) + 1/(1 + sinA) = 2sec^2A`
Prove that:
`(tanA + 1/cosA)^2 + (tanA - 1/cosA)^2 = 2((1 + sin^2A)/(1 - sin^2A))`
If 2 sin A – 1 = 0, show that: sin 3A = 3 sin A – 4 sin3 A
Prove the following identity :
`sec^4A - sec^2A = sin^2A/cos^4A`
If sec θ = `25/7`, then find the value of tan θ.
`(sin A)/(1 + cos A) + (1 + cos A)/(sin A)` = 2 cosec A
Prove that `sqrt(2 + tan^2 θ + cot^2 θ) = tan θ + cot θ`.
Prove that `(sec θ - 1)/(sec θ + 1) = ((sin θ)/(1 + cos θ ))^2`
Choose the correct alternative:
1 + cot2θ = ?
Choose the correct alternative:
tan (90 – θ) = ?
If cos A + cos2A = 1, then sin2A + sin4 A = ?
`sqrt((1 - cos^2theta) sec^2 theta) = tan theta`
Prove that (sec θ + tan θ) (1 – sin θ) = cos θ
Statement 1: sin2θ + cos2θ = 1
Statement 2: cosec2θ + cot2θ = 1
Which of the following is valid?
