Advertisements
Advertisements
Question
Prove the following trigonometric identities.
`(cot^2 A(sec A - 1))/(1 + sin A) = sec^2 A ((1 - sin A)/(1 + sec A))`
Advertisements
Solution
We have to prove `(cot^2 A(sec A - 1))/(1 + sin A) = sec^2 A ((1 - sin A)/(1 + sec A))`
We know that `sin^2 A = cos^2 A = 1`
`So,
`(cot^2 A(sec A - 1))/(1 + sin A) = sec^2 A ((1 - sin A)/(1 = sec A))`
`= (cos^2 A/sin^2 A(1/cos A - 1))/(1 + sin A)`
`= (cos^2 A/sin^2 A (1 - cos A)/(cos A))/(1 + sin A)`
`= (cos A(1 - cos A))/(sin^2 A(1 + sin A))`
`= (cos A (1 - cos A))/((1 - cos^2 A)(1 + sin A))`
`= (cos A (1 - cos A))/((1 - cos A)(1 + cos A)(1 + sin A))`
`= cos A/((1 + cos A)(1 + sin A))`
`= (1/sec A)/((1 + 1/sec A)(1 + sin A))`
`= (1/sec A)/(((sec A + 1)/sec A)) (1 + sin A)`
`= 1/((sec A +1)(1 + sin A))`
Multiplying both the numerator and denominator by (1 - sin A), we have
`= (1 - sin A)/((sec A + 1)(1 + sin A)(1 - sin A))`
`= (1 - sin A)/((sec A + 1)(1 - sin^2 A))`
`= (1 - sin A)/((sec A + 1)cos^2 A)`
`= sec^2 A ((1 - sin A))/((sec A + 1))`
`= sec^2 A ((1 - sin A)/(1 + sec A))`
Hence proved.
APPEARS IN
RELATED QUESTIONS
Prove the following identities:
`sqrt((1 + sinA)/(1 - sinA)) = sec A + tan A`
Prove the following identities:
`(cotA + cosecA - 1)/(cotA - cosecA + 1) = (1 + cosA)/sinA`
Prove the following identities:
`(1+ sin A)/(cosec A - cot A) - (1 - sin A)/(cosec A + cot A) = 2(1 + cot A)`
Prove the following identities:
`cosecA - cotA = sinA/(1 + cosA)`
`(tan A + tanB )/(cot A + cot B) = tan A tan B`
If tanθ `= 3/4` then find the value of secθ.
If sec θ + tan θ = x, write the value of sec θ − tan θ in terms of x.
If x = a sec θ and y = b tan θ, then b2x2 − a2y2 =
Prove the following identity :
`sin^4A + cos^4A = 1 - 2sin^2Acos^2A`
If sinA + cosA = `sqrt(2)` , prove that sinAcosA = `1/2`
Without using trigonometric table , evaluate :
`sin72^circ/cos18^circ - sec32^circ/(cosec58^circ)`
Prove that `sinA/sin(90^circ - A) + cosA/cos(90^circ - A) = sec(90^circ - A) cosec(90^circ - A)`
Prove that: 2(sin6θ + cos6θ) - 3 ( sin4θ + cos4θ) + 1 = 0.
Prove that `(sin (90° - θ))/cos θ + (tan (90° - θ))/cot θ + (cosec (90° - θ))/sec θ = 3`.
If tan A + sin A = m and tan A − sin A = n, then show that `m^2 - n^2 = 4 sqrt (mn)`.
Prove that `cot^2 "A" [(sec "A" - 1)/(1 + sin "A")] + sec^2 "A" [(sin"A" - 1)/(1 + sec"A")]` = 0
If sec θ = `41/40`, then find values of sin θ, cot θ, cosec θ
If tan θ = 3, then `(4 sin theta - cos theta)/(4 sin theta + cos theta)` is equal to ______.
Prove the following:
`1 + (cot^2 alpha)/(1 + "cosec" alpha)` = cosec α
Prove that `sqrt(sec^2 theta + "cosec"^2 theta) = tan theta + cot theta`
