Advertisements
Advertisements
प्रश्न
Prove that `sintheta/(sectheta+ 1) +sintheta/(sectheta - 1)` = 2 cot θ
Advertisements
उत्तर
L.H.S = `sintheta/(sectheta+ 1) +sintheta/(sectheta - 1)`
= `sintheta/(1/costheta + 1) + sintheta/(1/costheta - 1`
= `sintheta/((1 + costheta)/costheta) + sintheta/((1 - costheta)/(costheta))`
= `(sintheta costheta)/(1 + costheta) + (sintheta costheta)/(1 - costheta)`
= `sin theta costheta (1 /(1 + costheta) + 1/(1 - costheta))`
= `sintheta costheta [(1 - costheta + 1 + costheta)/((1 + costheta)(1 - costheta))]`
= `sintheta costheta (2/(1 - cos^2theta))` ......[∵ (a + b)(a – b) = a2 – b2]
= `sintheta costheta xx 2/(sin^2theta)` .....`[(because sin^2theta + cos^2theta = 1),(therefore 1 - cos^2theta = sin^2theta)]`
= `2 xx (costheta)/(sintheta)`
= 2cot θ
= R.H.S
∴ `sintheta/(sectheta+ 1) +sintheta/(sectheta - 1)` = 2 cot θ
APPEARS IN
संबंधित प्रश्न
Prove the following trigonometric identities.
`tan theta + 1/tan theta` = sec θ.cosec θ
Prove that `sqrt((1 + cos theta)/(1 - cos theta)) + sqrt((1 - cos theta)/(1 + cos theta)) = 2 cosec theta`
Prove the following identities:
(1 – tan A)2 + (1 + tan A)2 = 2 sec2A
Prove the following identities:
`1/(1 - sinA) + 1/(1 + sinA) = 2sec^2A`
Prove that:
`(cosecA - sinA)(secA - cosA) = 1/(tanA + cotA)`
If x = r sin A cos B, y = r sin A sin B and z = r cos A, then prove that : x2 + y2 + z2 = r2
`(1-cos^2theta) sec^2 theta = tan^2 theta`
`(cos theta cosec theta - sin theta sec theta )/(costheta + sin theta) = cosec theta - sec theta`
`(tan A + tanB )/(cot A + cot B) = tan A tan B`
Prove that:
`"tan A"/(1 + "tan"^2 "A")^2 + "Cot A"/(1 + "Cot"^2 "A")^2 = "sin A cos A"`.
Prove the following identity :
`sin^4A + cos^4A = 1 - 2sin^2Acos^2A`
Prove the following identity :
`((1 + tan^2A)cotA)/(cosec^2A) = tanA`
Prove the following identity :
`(sinA + cosA)/(sinA - cosA) + (sinA - cosA)/(sinA + cosA) = 2/(2sin^2A - 1)`
Prove that:
`sqrt((sectheta - 1)/(sec theta + 1)) + sqrt((sectheta + 1)/(sectheta - 1)) = 2cosectheta`
Prove that `sqrt((1 + cos A)/(1 - cos A)) = (tan A + sin A)/(tan A. sin A)`
Without using a trigonometric table, prove that
`(cos 70°)/(sin 20°) + (cos 59°)/(sin 31°) - 8sin^2 30° = 0`.
If `sqrt(3)` sin θ – cos θ = θ, then show that tan 3θ = `(3tan theta - tan^3 theta)/(1 - 3 tan^2 theta)`
Prove that sin4A – cos4A = 1 – 2cos2A
Prove that sin6A + cos6A = 1 – 3sin2A . cos2A
Prove the following:
`tanA/(1 + sec A) - tanA/(1 - sec A)` = 2cosec A
