Advertisements
Advertisements
Question
Prove the following identities.
`sqrt((1 + sin theta)/(1 - sin theta)) + sqrt((1 - sin theta)/(1 + sin theta))` = 2 sec θ
Advertisements
Solution
`sqrt((1 + sin theta)/(1 - sin theta)) + sqrt((1 - sin theta)/(1 + sin theta))` = 2 sec θ
`sqrt((1 + sin theta)/(1 - sin theta)) = sqrt(((1 + sin theta)(1 + sin theta))/((1 - sin theta)(1 + sin theta))`
= `sqrt((1 + sin theta)^2/(1 - sin^2 theta)`
= `sqrt((1 + sin theta)^2/(cos^2 theta)`
= `(1 + sin theta)/cos theta`
`sqrt(((1 - sin theta))/((1 + sin theta))) = sqrt(((1 - sin theta))/((1 - sin theta)) xx ((1 + sin theta))/((1 - sin theta))`
= `sqrt((1 - sin theta)^2/(1 - sin^2 theta)`
= `sqrt((1- sin theta)^2/(cos^2 theta)) = (1 - sin theta)/cos theta`
L.H.S. = `sqrt((1 + sin theta)/(1 - sin theta)) + sqrt((1 - sin theta)/(1 + sin theta)`
= `(1 + sin theta)/cos theta + (1 - sin theta)/cos theta`
= `(1 + sin theta + 1 - sin theta)/cos theta`
= `2/cos theta`
= 2 sec θ
L.H.S. = R.H.S.
APPEARS IN
RELATED QUESTIONS
Prove the following identities:
`(i) cos4^4 A – cos^2 A = sin^4 A – sin^2 A`
`(ii) cot^4 A – 1 = cosec^4 A – 2cosec^2 A`
`(iii) sin^6 A + cos^6 A = 1 – 3sin^2 A cos^2 A.`
If tanθ + sinθ = m and tanθ – sinθ = n, show that `m^2 – n^2 = 4\sqrt{mn}.`
`"If "\frac{\cos \alpha }{\cos \beta }=m\text{ and }\frac{\cos \alpha }{\sin \beta }=n " show that " (m^2 + n^2 ) cos^2 β = n^2`
Prove the following identities:
`sqrt((1 + sinA)/(1 - sinA)) = sec A + tan A`
Prove the following identities:
`sqrt((1 - sinA)/(1 + sinA)) = cosA/(1 + sinA)`
2 (sin6 θ + cos6 θ) − 3 (sin4 θ + cos4 θ) is equal to
If a cos θ − b sin θ = c, then a sin θ + b cos θ =
Prove the following identity :
`(cos^3A + sin^3A)/(cosA + sinA) + (cos^3A - sin^3A)/(cosA - sinA) = 2`
Prove that tan2Φ + cot2Φ + 2 = sec2Φ.cosec2Φ.
Prove that `(cot A)/(1 - cot A) + (tan A)/(1 - tan A) = -1`.
