Advertisements
Advertisements
Question
Prove that `"cot A"/(1 - tan "A") + "tan A"/(1 - cot"A")` = 1 + tan A + cot A = sec A . cosec A + 1
Advertisements
Solution
`"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
RELATED QUESTIONS
Evaluate
`(sin ^2 63^@ + sin^2 27^@)/(cos^2 17^@+cos^2 73^@)`
Prove the following identities, where the angles involved are acute angles for which the expressions are defined:
`(cosec θ – cot θ)^2 = (1-cos theta)/(1 + cos theta)`
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.
`sqrt((1 - cos theta)/(1 + cos theta)) = cosec theta - cot theta`
Prove the following trigonometric identities.
`(cos^2 theta)/sin theta - cosec theta + sin theta = 0`
Prove the following trigonometric identities
tan2 A + cot2 A = sec2 A cosec2 A − 2
If sin θ + cos θ = x, prove that `sin^6 theta + cos^6 theta = (4- 3(x^2 - 1)^2)/4`
Show that : `sinA/sin(90^circ - A) + cosA/cos(90^circ - A) = sec A cosec A`
Prove the following identities:
`(sinA - cosA + 1)/(sinA + cosA - 1) = cosA/(1 - sinA)`
`(tan theta)/((sec theta -1))+(tan theta)/((sec theta +1)) = 2 sec theta`
Prove the following identity :
`(cosecA)/(cosecA - 1) + (cosecA)/(cosecA + 1) = 2sec^2A`
If x = acosθ , y = bcotθ , prove that `a^2/x^2 - b^2/y^2 = 1.`
Prove that `( tan A + sec A - 1)/(tan A - sec A + 1) = (1 + sin A)/cos A`.
Prove the following identities:
`1/(sin θ + cos θ) + 1/(sin θ - cos θ) = (2sin θ)/(1 - 2 cos^2 θ)`.
Prove that identity:
`(sec A - 1)/(sec A + 1) = (1 - cos A)/(1 + cos A)`
If sec θ = `25/7`, find the value of tan θ.
Solution:
1 + tan2 θ = sec2 θ
∴ 1 + tan2 θ = `(25/7)^square`
∴ tan2 θ = `625/49 - square`
= `(625 - 49)/49`
= `square/49`
∴ tan θ = `square/7` ........(by taking square roots)
Prove the following:
`1 + (cot^2 alpha)/(1 + "cosec" alpha)` = cosec α
Prove that `(1 + sec theta - tan theta)/(1 + sec theta + tan theta) = (1 - sin theta)/cos theta`
Statement 1: sin2θ + cos2θ = 1
Statement 2: cosec2θ + cot2θ = 1
Which of the following is valid?
