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
Prove the following trigonometric identities
`cos theta/(1 - sin theta) = (1 + sin theta)/cos theta`
Prove the following trigonometric identities.
sec6θ = tan6θ + 3 tan2θ sec2θ + 1
Prove the following trigonometric identities
cosec6θ = cot6θ + 3 cot2θ cosec2θ + 1
Prove the following trigonometric identities.
`(cosec A)/(cosec A - 1) + (cosec A)/(cosec A = 1) = 2 sec^2 A`
Prove the following trigonometric identities.
`sin A/(sec A + tan A - 1) + cos A/(cosec A + cot A + 1) = 1`
Prove the following identities:
(sin A + cosec A)2 + (cos A + sec A)2 = 7 + tan2 A + cot2 A
If m = a sec A + b tan A and n = a tan A + b sec A, then prove that : m2 – n2 = a2 – b2
If sin A + cos A = p and sec A + cosec A = q, then prove that : q(p2 – 1) = 2p.
`sin theta / ((1+costheta))+((1+costheta))/sin theta=2cosectheta`
If x=a `cos^3 theta and y = b sin ^3 theta ," prove that " (x/a)^(2/3) + ( y/b)^(2/3) = 1.`
Write the value of ` sec^2 theta ( 1+ sintheta )(1- sintheta).`
If ` cot A= 4/3 and (A+ B) = 90° ` ,what is the value of tan B?
Find the value of `(cos 38° cosec 52°)/(tan 18° tan 35° tan 60° tan 72° tan 55°)`
\[\frac{1 - \sin \theta}{\cos \theta}\] is equal to
Prove the following identity :
`sqrt((1 + cosA)/(1 - cosA)) = cosecA + cotA`
Prove the following identity :
`(cotA + cosecA - 1)/(cotA - cosecA + 1) = (cosA + 1)/sinA`
Prove the following identity :
`(sec^2θ - sin^2θ)/tan^2θ = cosec^2θ - cos^2θ`
Without using trigonometric table , evaluate :
`sin72^circ/cos18^circ - sec32^circ/(cosec58^circ)`
Without using trigonometric identity , show that :
`tan10^circ tan20^circ tan30^circ tan70^circ tan80^circ = 1/sqrt(3)`
Prove that sin4θ - cos4θ = sin2θ - cos2θ
= 2sin2θ - 1
= 1 - 2 cos2θ
