Advertisements
Advertisements
Question
Prove that `(1 + sin θ)/(1 - sin θ) = (sec θ + tan θ)^2`.
Advertisements
Solution
L.H.S. = `(1 + sin θ)/(1 - sin θ)`
= `((1 + sinθ)/(cosθ))/((1 - sinθ)/(cosθ))` ...[Dividing numerator and denominator by cos θ]
= `(1/cosθ + (sinθ)/(cosθ))/(1/cosθ - (sinθ)/(cosθ)`
= `(secθ + tanθ)/(secθ - tanθ)`
= `(secθ + tanθ)/(secθ - tanθ) xx (secθ + tanθ)/(secθ + tanθ)` ...[On rationalising the denominator]
= `(secθ + tanθ)^2/(sec^2θ - tan^2θ)`
= `(secθ + tanθ)^2/1` ...`[(∵ 1 + tan^2θ = sec^2θ),(∴ sec^2θ - tan^2θ = 1)]`
= (sec θ + tan θ)2
= R.H.S.
∴ `(1 + sinθ)/(1 - sinθ) = (sec θ + tan θ)^2`
APPEARS IN
RELATED QUESTIONS
Prove the following trigonometric identities.
`(cos theta)/(cosec theta + 1) + (cos theta)/(cosec theta - 1) = 2 tan theta`
Prove that:
`tanA/(1 - cotA) + cotA/(1 - tanA) = secA "cosec" A + 1`
Show that : tan 10° tan 15° tan 75° tan 80° = 1
`tan theta/(1+ tan^2 theta)^2 + cottheta/(1+ cot^2 theta)^2 = sin theta cos theta`
`sqrt((1-cos theta)/(1+cos theta)) = (cosec theta - cot theta)`
Show that none of the following is an identity:
`tan^2 theta + sin theta = cos^2 theta`
Write the value of `cosec^2 theta (1+ cos theta ) (1- cos theta).`
What is the value of \[\frac{\tan^2 \theta - \sec^2 \theta}{\cot^2 \theta - {cosec}^2 \theta}\]
What is the value of (1 + tan2 θ) (1 − sin θ) (1 + sin θ)?
If \[\sin \theta = \frac{1}{3}\] then find the value of 2cot2 θ + 2.
Prove the following identity :
`tanA - cotA = (1 - 2cos^2A)/(sinAcosA)`
Prove the following identity :
`(secA - 1)/(secA + 1) = sin^2A/(1 + cosA)^2`
Prove the following identity :
`(1 + tan^2A) + (1 + 1/tan^2A) = 1/(sin^2A - sin^4A)`
Prove the following identity :
`(1 + tan^2θ)sinθcosθ = tanθ`
Find the value of `θ(0^circ < θ < 90^circ)` if :
`tan35^circ cot(90^circ - θ) = 1`
There are two poles, one each on either bank of a river just opposite to each other. One pole is 60 m high. From the top of this pole, the angle of depression of the top and foot of the other pole are 30° and 60° respectively. Find the width of the river and height of the other pole.
Prove the following identities.
sec6 θ = tan6 θ + 3 tan2 θ sec2 θ + 1
cot θ . tan θ = ?
Prove that cos2θ . (1 + tan2θ) = 1. Complete the activity given below.
Activity:
L.H.S. = `square`
= `cos^2θ xx square` ...`[1 + tan^2θ = square]`
= `(cos θ xx square)^2`
= 12
= 1
= R.H.S.
