Advertisements
Advertisements
Question
Show that: `tan "A"/(1 + tan^2 "A")^2 + cot "A"/(1 + cot^2 "A")^2 = sin"A" xx cos"A"`
Advertisements
Solution
Proof: L.H.S. = `tan"A"/(1 + tan^2 "A")^2 + cot"A"/(1 + cot^2 "A")^2`
= `tan "A"/(sec^2"A")^2 + cot "A"/("cosec"^2"A")^2` ......`[(∵ 1 + cot^2θ = "cosec"^2θ),(1 + tan^2θ = sec^2θ)]`
= `tan "A"/sec^4"A" + cot "A"/("cosec"^4"A")`
= `sin "A"/cos "A" xx 1/(sec^4 "A") + cos "A"/sin "A" xx 1/("cosec"^4 "A")`
= `sin "A"/cos "A" xx cos^4"A" + cos "A"/sin "A" xx sin^4"A"`
= sinA × cos3A + cosA × sin3A
= sinA cosA (cos2A + sin2A)
= sinA cosA (1) ......[∵ cos2A + sin2A = 1]
= sinA.cosA
= R.H.S
L.H.S. = R.H.S.
Hence proved.
APPEARS IN
RELATED QUESTIONS
Prove the following trigonometric identities
`(1 + tan^2 theta)/(1 + cot^2 theta) = ((1 - tan theta)/(1 - cot theta))^2 = tan^2 theta`
Prove the following trigonometric identities.
`sqrt((1 - cos A)/(1 + cos A)) = cosec A - cot A`
Prove the following identities:
`cosecA + cotA = 1/(cosecA - cotA)`
If tan A = n tan B and sin A = m sin B, prove that `cos^2A = (m^2 - 1)/(n^2 - 1)`
`(1+ cos theta - sin^2 theta )/(sin theta (1+ cos theta))= cot theta`
If `tan theta = 1/sqrt(5), "write the value of" (( cosec^2 theta - sec^2 theta))/(( cosec^2 theta - sec^2 theta))`.
If ` cot A= 4/3 and (A+ B) = 90° ` ,what is the value of tan B?
If `sin theta = x , " write the value of cot "theta .`
If sec2 θ (1 + sin θ) (1 − sin θ) = k, then find the value of k.
If sin θ + sin2 θ = 1, then cos2 θ + cos4 θ =
Prove the following identity :
`sec^2A.cosec^2A = tan^2A + cot^2A + 2`
Prove the following identity :
`sqrt((1 - cosA)/(1 + cosA)) = sinA/(1 + cosA)`
Prove the following identity :
`tan^2A - tan^2B = (sin^2A - sin^2B)/(cos^2Acos^2B)`
Prove the following identity :
`(sinA - sinB)/(cosA + cosB) + (cosA - cosB)/(sinA + sinB) = 0`
Without using trigonometric table , evaluate :
`(sin47^circ/cos43^circ)^2 - 4cos^2 45^circ + (cos43^circ/sin47^circ)^2`
For ΔABC , prove that :
`tan ((B + C)/2) = cot "A/2`
Prove that:
`sqrt((sectheta - 1)/(sec theta + 1)) + sqrt((sectheta + 1)/(sectheta - 1)) = 2cosectheta`
Prove that: 2(sin6θ + cos6θ) - 3 ( sin4θ + cos4θ) + 1 = 0.
Choose the correct alternative:
Which is not correct formula?
If sin θ + cos θ = p and sec θ + cosec θ = q, then prove that q(p2 – 1) = 2p.
