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प्रश्न
`tan theta /((1 - cot theta )) + cot theta /((1 - tan theta)) = (1+ sec theta cosec theta)`
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
APPEARS IN
संबंधित प्रश्न
Prove the following trigonometric identities:
(i) (1 – sin2θ) sec2θ = 1
(ii) cos2θ (1 + tan2θ) = 1
`(1+tan^2A)/(1+cot^2A)` = ______.
Prove the following trigonometric identities.
`(cos theta - sin theta + 1)/(cos theta + sin theta - 1) = cosec theta + cot theta`
Prove the following identities:
`tan A - cot A = (1 - 2cos^2A)/(sin A cos A)`
Prove that:
(sec A − tan A)2 (1 + sin A) = (1 − sin A)
`(sin theta)/((sec theta + tan theta -1)) + cos theta/((cosec theta + cot theta -1))=1`
If `( sin theta + cos theta ) = sqrt(2) , " prove that " cot theta = ( sqrt(2)+1)`.
Prove the following identity :
cosecθ(1 + cosθ)(cosecθ - cotθ) = 1
Prove the following identity :
`tan^2A - sin^2A = tan^2A.sin^2A`
Prove the following identity :
`sinA/(1 + cosA) + (1 + cosA)/sinA = 2cosecA`
Prove the following identity :
`sqrt((1 + sinq)/(1 - sinq)) + sqrt((1- sinq)/(1 + sinq))` = 2secq
Prove the following identity :
`sin^8θ - cos^8θ = (sin^2θ - cos^2θ)(1 - 2sin^2θcos^2θ)`
If m = a secA + b tanA and n = a tanA + b secA , prove that m2 - n2 = a2 - b2
Without using trigonometric identity , show that :
`sin42^circ sec48^circ + cos42^circ cosec48^circ = 2`
Prove that:
`sqrt((sectheta - 1)/(sec theta + 1)) + sqrt((sectheta + 1)/(sectheta - 1)) = 2cosectheta`
Prove the following identities.
`(1 - tan^2theta)/(cot^2 theta - 1)` = tan2 θ
If 1 – cos2θ = `1/4`, then θ = ?
If tan θ = `7/24`, then to find value of cos θ complete the activity given below.
Activity:
sec2θ = 1 + `square` ......[Fundamental tri. identity]
sec2θ = 1 + `square^2`
sec2θ = 1 + `square/576`
sec2θ = `square/576`
sec θ = `square`
cos θ = `square` .......`[cos theta = 1/sectheta]`
If a sinθ + b cosθ = c, then prove that a cosθ – b sinθ = `sqrt(a^2 + b^2 - c^2)`.
(sec2 θ – 1) (cosec2 θ – 1) is equal to ______.
