Advertisements
Advertisements
प्रश्न
If cos A = `(2sqrt("m"))/("m" + 1)`, then prove that cosec A = `("m" + 1)/("m" - 1)`
उत्तर
cos A = `(2sqrt("m"))/("m" + 1)` ......[Given]
We know that,
sin2A + cos2A = 1
∴ `sin^2"A" + ((2sqrt("m"))/("m" + 1))^2` = 1
∴ `sin^2"A" + (4"m")/("m" + 1)^2` = 1
∴ sin2A = `1 - (4"m")/("m" + 1)^2`
= `(("m" + 1)^2 - 4"m")/("m" + 1)^2`
= `("m"^2 + 2"m" + 1 - 4"m")/("m" + 1)^2` ......[∵ (a + b)2 = a2 + 2ab + b2]
= `("m"^2 - 2"m" + 1)/("m" + 1)^2`
∴ sin2A = `("m" - 1)^2/("m" + 1)^2` ......[∵ a2 – 2ab + b2 = (a – b)2]
∴ sin A = `("m" - 1)/("m" + 1)` .....[Taking square root of both sides]
Now, cosec A = `1/"sin A"`
= `1/(("m" - 1)/("m" + 1))`
∴ cosec A = `("m" + 1)/("m" - 1)`
APPEARS IN
संबंधित प्रश्न
Prove that:
sec2θ + cosec2θ = sec2θ x cosec2θ
Prove the following trigonometric identities.
`(1 + cos theta + sin theta)/(1 + cos theta - sin theta) = (1 + sin theta)/cos theta`
If sin A + cos A = p and sec A + cosec A = q, then prove that : q(p2 – 1) = 2p.
`1+(tan^2 theta)/((1+ sec theta))= sec theta`
`(1+tan^2theta)(1+cot^2 theta)=1/((sin^2 theta- sin^4theta))`
`sqrt((1+cos theta)/(1-cos theta)) + sqrt((1-cos theta )/(1+ cos theta )) = 2 cosec theta`
`(cos theta cosec theta - sin theta sec theta )/(costheta + sin theta) = cosec theta - sec theta`
`(1+ tan theta + cot theta )(sintheta - cos theta) = ((sec theta)/ (cosec^2 theta)-( cosec theta)/(sec^2 theta))`
If `(cot theta ) = m and ( sec theta - cos theta) = n " prove that " (m^2 n)(2/3) - (mn^2)(2/3)=1`
Write the value of `(1 - cos^2 theta ) cosec^2 theta`.
Prove the following identity :
`cosec^4A - cosec^2A = cot^4A + cot^2A`
Prove the following identity :
`1/(tanA + cotA) = sinAcosA`
Prove the following identity :
`(cosecA)/(cosecA - 1) + (cosecA)/(cosecA + 1) = 2sec^2A`
Prove the following identity :
`sqrt((1 - cosA)/(1 + cosA)) = sinA/(1 + cosA)`
Prove that: (1+cot A - cosecA)(1 + tan A+ secA) =2.
Prove that: `1/(sec θ - tan θ) = sec θ + tan θ`.
Prove that: `(sin A + cos A)/(sin A - cos A) + (sin A - cos A)/(sin A + cos A) = 2/(sin^2 A - cos^2 A)`.
If sin θ + cos θ = a and sec θ + cosec θ = b , then the value of b(a2 – 1) is equal to
If 4 tanβ = 3, then `(4sinbeta-3cosbeta)/(4sinbeta+3cosbeta)=` ______.
If 5 tan β = 4, then `(5 sin β - 2 cos β)/(5 sin β + 2 cos β)` = ______.
