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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 that `sqrt(sec^2 theta + cosec^2 theta) = tan theta + cot theta`
Prove the following trigonometric identities.
`sin^2 A + 1/(1 + tan^2 A) = 1`
Prove the following trigonometric identities.
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If 3 sin θ + 5 cos θ = 5, prove that 5 sin θ – 3 cos θ = ± 3.
If x = a sec θ cos ϕ, y = b sec θ sin ϕ and z = c tan θ, show that `x^2/a^2 + y^2/b^2 - x^2/c^2 = 1`
Prove the following identities:
`(1 - 2sin^2A)^2/(cos^4A - sin^4A) = 2cos^2A - 1`
If sin A + cos A = p and sec A + cosec A = q, then prove that : q(p2 – 1) = 2p.
Prove that secθ + tanθ =`(costheta)/(1-sintheta)`.
If a cot θ + b cosec θ = p and b cot θ − a cosec θ = q, then p2 − q2
If a cos θ + b sin θ = m and a sin θ − b cos θ = n, then a2 + b2 =
Prove the following identity :
`(1 - tanA)^2 + (1 + tanA)^2 = 2sec^2A`
Prove the following identity :
`sinA/(1 + cosA) + (1 + cosA)/sinA = 2cosecA`
If `x/(a cosθ) = y/(b sinθ) "and" (ax)/cosθ - (by)/sinθ = a^2 - b^2 , "prove that" x^2/a^2 + y^2/b^2 = 1`
If A = 60°, B = 30° verify that tan( A - B) = `(tan A - tan B)/(1 + tan A. tan B)`.
Prove that `sin A/(sec A + tan A - 1) + cos A/(cosec A + cot A - 1) = 1`.
If x = h + a cos θ, y = k + b sin θ.
Prove that `((x - h)/a)^2 + ((y - k)/b)^2 = 1`.
Prove the following identities:
`(1 - tan^2 θ)/(cot^2 θ - 1) = tan^2 θ`.
Show that tan 7° × tan 23° × tan 60° × tan 67° × tan 83° = `sqrt(3)`.
Show that, cotθ + tanθ = cosecθ × secθ
Solution :
L.H.S. = cotθ + tanθ
= `cosθ/sinθ + sinθ/cosθ`
= `(square + square)/(sinθ xx cosθ)`
= `1/(sinθ xx cosθ)` ............... `square`
= `1/sinθ xx 1/square`
= cosecθ × secθ
L.H.S. = R.H.S
∴ cotθ + tanθ = cosecθ × secθ
Prove the following trigonometry identity:
(sin θ + cos θ)(cosec θ – sec θ) = cosec θ ⋅ sec θ – 2 tan θ
