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, where the angles involved are acute angles for which the expressions are defined:
`(1+ secA)/sec A = (sin^2A)/(1-cosA)`
[Hint : Simplify LHS and RHS separately.]
Prove the following trigonometric identities.
`((1 + sin theta - cos theta)/(1 + sin theta + cos theta))^2 = (1 - cos theta)/(1 + cos theta)`
If `cosA/cosB = m` and `cosA/sinB = n`, show that : (m2 + n2) cos2 B = n2.
`(tan theta)/((sec theta -1))+(tan theta)/((sec theta +1)) = 2 sec theta`
The value of \[\sqrt{\frac{1 + \cos \theta}{1 - \cos \theta}}\]
If A + B = 90°, show that `(sin B + cos A)/sin A = 2tan B + tan A.`
Prove the following identities.
sec6 θ = tan6 θ + 3 tan2 θ sec2 θ + 1
cos θ . sec θ = ?
If 5 sec θ – 12 cosec θ = 0, then find values of sin θ, sec θ.
If tan θ = 3, then `(4 sin theta - cos theta)/(4 sin theta + cos theta)` is equal to ______.
