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प्रश्न
`(cos ec^theta + cot theta )/( cos ec theta - cot theta ) = (cosec theta + cot theta )^2 = 1+2 cot^2 theta + 2cosec theta cot theta`
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उत्तर
Here, `( cosec theta + cot theta )/( cosec theta - cot theta)`
= `((cosec theta + cot theta) ( cosec theta + cot theta ))/(( cosec theta - cot theta ) ( cosec theta + cot theta))`
=` ((cosec theta + cot theta)^2)/(( cosec ^2 theta - cot^2 theta))`
=`((cosec theta + cot theta )^2) /1`
=`(cosec theta + cot theta )^2`
Again , `( cosec theta + cot theta )^2`
= ` cosec^2 theta + cot^2 theta + 2 cosec theta cot theta `
=` 1+cot^2 theta + cot^2 theta + 2 cosec theta cot theta (∵ cosec^2 theta - cot^2 theta =1)`
=` 1+2 cot^2 theta + 2 cosec theta cot theta `
APPEARS IN
संबंधित प्रश्न
Prove the following trigonometric identities.
`"cosec" theta sqrt(1 - cos^2 theta) = 1`
Prove the following trigonometric identities.
(sec2 θ − 1) (cosec2 θ − 1) = 1
Prove the following trigonometric identities. `(1 - cos A)/(1 + cos A) = (cot A - cosec A)^2`
Prove the following trigonometric identities.
`1/(sec A - 1) + 1/(sec A + 1) = 2 cosec A cot A`
Prove that: `sqrt((sec theta - 1)/(sec theta + 1)) + sqrt((sec theta + 1)/(sec theta - 1)) = 2 cosec theta`
Prove the following identities:
`(secA - tanA)/(secA + tanA) = 1 - 2secAtanA + 2tan^2A`
Prove that:
(sin A + cos A) (sec A + cosec A) = 2 + sec A cosec A
If x= a sec `theta + b tan theta and y = a tan theta + b sec theta ,"prove that" (x^2 - y^2 )=(a^2 -b^2)`
Find the value of sin 30° + cos 60°.
Prove that sec θ. cosec (90° - θ) - tan θ. cot( 90° - θ ) = 1.
Prove that `sqrt((1 + sin θ)/(1 - sin θ))` = sec θ + tan θ.
Prove that `( tan A + sec A - 1)/(tan A - sec A + 1) = (1 + sin A)/cos A`.
If `tan θ = 9/40`, complete the activity to find the value of sec θ.
Activity:
sec2θ = 1 + `square` ...[Fundamental trigonometric identity]
sec2θ = 1 + `square^2`
sec2θ = 1 + `square`
sec θ = `square`
Prove that sin2A . tan A + cos2A . cot A + 2 sin A . cos A = tan A + cot A.
If tan θ – sin2θ = cos2θ, then show that `sin^2θ = 1/2`.
Prove that `sqrt(sec^2 theta + "cosec"^2 theta) = tan theta + cot theta`
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.
`(cos^2 θ)/(sin^2 θ) - 1/(sin^2 θ)`, in simplified form, is ______.
If tan θ = `x/y`, then cos θ is equal to ______.
