Advertisements
Advertisements
प्रश्न
If cos A = `(2sqrt(m))/(m + 1)`, then prove that cosec A = `(m + 1)/(m - 1)`.
Advertisements
उत्तर
`cos A = (2sqrt(m))/(m + 1)` ...[Given]
We know that,
sin2A + cos2A = 1
∴ `sin^2A + ((2sqrt(m))/(m + 1))^2 = 1`
∴ `sin^2A + (4m)/(m + 1)^2 = 1`
∴ `sin^2A = 1 - (4m)/(m + 1)^2`
= `((m + 1)^2 - 4m)/(m + 1)^2`
= `(m^2 + 2m + 1 - 4m)/(m + 1)^2` ...[∵ (a + b)2 = a2 + 2ab + b2]
= `(m^2 - 2m + 1)/(m + 1)^2`
∴ `sin^2A = (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 the following trigonometric identities:
`(\text{i})\text{ }\frac{\sin \theta }{1-\cos \theta }=\text{cosec}\theta+\cot \theta `
If (secA + tanA)(secB + tanB)(secC + tanC) = (secA – tanA)(secB – tanB)(secC – tanC) prove that each of the side is equal to ±1. We have,
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
`(sintheta - 2sin^3theta)/(2costheta - costheta) =tan theta`
Prove the following trigonometric identities.
`(tan^3 theta)/(1 + tan^2 theta) + (cot^3 theta)/(1 + cot^2 theta) = sec theta cosec theta - 2 sin theta cos theta`
Prove the following trigonometric identities.
if x = a cos^3 theta, y = b sin^3 theta` " prove that " `(x/a)^(2/3) + (y/b)^(2/3) = 1`
Prove the following identities:
`1/(1 - sinA) + 1/(1 + sinA) = 2sec^2A`
Prove the following identities:
`(sintheta - 2sin^3theta)/(2cos^3theta - costheta) = tantheta`
Prove the following identities:
(1 + cot A – cosec A)(1 + tan A + sec A) = 2
Prove the following identities:
`(costhetacottheta)/(1 + sintheta) = cosectheta - 1`
Prove that:
`(cosecA - sinA)(secA - cosA) = 1/(tanA + cotA)`
Prove that:
(cosec A – sin A) (sec A – cos A) sec2 A = tan A
`sec theta (1- sin theta )( sec theta + tan theta )=1`
`tan theta/(1+ tan^2 theta)^2 + cottheta/(1+ cot^2 theta)^2 = sin theta cos theta`
If \[\sin \theta = \frac{4}{5}\] what is the value of cotθ + cosecθ?
\[\frac{1 + \tan^2 A}{1 + \cot^2 A}\]is equal to
Prove the following identity :
`(cosecθ)/(tanθ + cotθ) = cosθ`
Prove that sec2 (90° - θ) + tan2 (90° - θ) = 1 + 2 cot2 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 `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 the following:
`1 + (cot^2 alpha)/(1 + "cosec" alpha)` = cosec α
