Advertisements
Advertisements
प्रश्न
Prove that `"cot A"/(1 - tan "A") + "tan A"/(1 - cot"A")` = 1 + tan A + cot A = sec A . cosec A + 1
उत्तर
`"cot A"/(1 - tan "A") + "tan A"/(1 - cot"A")`
= `((cos "A")/(sin "A"))/(1 - (sin "A")/(cos "A")) + ((sin "A")/(cos "A"))/(1 - (cos "A")/(sin "A"))`
= `((cos "A")/(sin "A"))/((cos "A" - sin "A")/(cos "A")) + ((sin "A")/(cos "A"))/((sin "A" - cos "A")/(sin "A"))`
= `"cos A"/"sin A" xx "cos A"/(cos "A" - sin "A") + "sin A"/"cos A" xx "sin A"/(sin "A" - cos "A")`
= `(cos^2"A")/(sin "A"(cos "A" - sin "A")) + (sin^2"A")/(cos"A"(sin"A" - cos"A"))`
= `1/(sin "A" - cos "A") ((-cos^3"A" + sin^3"A")/(sin"A" cos"A"))`
= `1/(sin"A" - cos"A")((sin^3"A" - cos^3"A")/(sin"A" cos"A"))`
= `1/(sin"A" - cos"A")xx ((sin"A" - cos"A")(sin^2"A" + sin"A" cos"A" + cos^2"A"))/(sin"A" cos"A")` ......[∵ a3 – b3 = (a – b)(a2 + ab + b2)]
= `(sin^2"A" +sin"A" cos"A" + cos^2"A")/(sin"A" cos"A"` ......(i)
= `(1 + sin"A" cos"A")/(sin"A" cos"A")` .....[∵ sin2A + cos2A = 1]
= `1/(sin"A" cos"A") + (sin"A" cos"A")/(sin"A" cos"A")`
= cosec A sec A + 1 .....(ii)
`"cot A"/(1 - tan "A") + "tan A"/(1 - cot "A")`
= `(sin^2"A" + sin"A" cos"A" + cos^2"A")/(sin"A" cos"A")` ......[From (i)]
= `(sin^2"A")/(sin"A" cos"A") + "sin A cos A"/"sin A cos A" + (cos^2"A")/"sin A cos A"`
= `"sin A"/"cos A" + 1 + "cos A"/"sin A"`
= tan A + 1 + cot A ......(iii)
From (ii) and (iii), we get
`"cot A"/(1 - tan "A") + "tan A"/(1 - cot "A")` = 1 + tan A + cot A = sec A . cosec A + 1
संबंधित प्रश्न
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
`(sintheta - 2sin^3theta)/(2costheta - costheta) =tan theta`
if `cos theta = 5/13` where `theta` is an acute angle. Find the value of `sin 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`
If cos θ + cot θ = m and cosec θ – cot θ = n, prove that mn = 1
Prove the following identities:
(cos A + sin A)2 + (cos A – sin A)2 = 2
Prove the following identities:
sec2 A . cosec2 A = tan2 A + cot2 A + 2
Prove that:
`tanA/(1 - cotA) + cotA/(1 - tanA) = secA cosecA + 1`
Prove that:
(sin A + cos A) (sec A + cosec A) = 2 + sec A cosec A
If` (sec theta + tan theta)= m and ( sec theta - tan theta ) = n ,` show that mn =1
Write True' or False' and justify your answer the following :
The value of sin θ+cos θ is always greater than 1 .
Prove the following identity :
`sinA/(1 + cosA) + (1 + cosA)/sinA = 2cosecA`
Prove the following identity :
`(1 + cotA)^2 + (1 - cotA)^2 = 2cosec^2A`
If `asin^2θ + bcos^2θ = c and p sin^2θ + qcos^2θ = r` , prove that (b - c)(r - p) = (c - a)(q - r)
Prove that: 2(sin6θ + cos6θ) - 3 ( sin4θ + cos4θ) + 1 = 0.
Prove that:
`(cos^3 θ + sin^3 θ)/(cos θ + sin θ) + (cos^3 θ - sin^3 θ)/(cos θ - sin θ) = 2`
Prove that sin2 5° + sin2 10° .......... + sin2 85° + sin2 90° = `9 1/2`.
Prove that sin6A + cos6A = 1 – 3sin2A . cos2A
If sin θ + cos θ = `sqrt(3)`, then show that tan θ + cot θ = 1
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.
If 2 cos θ + sin θ = `1(θ ≠ π/2)`, then 7 cos θ + 6 sin θ is equal to ______.
