Advertisements
Advertisements
प्रश्न
Prove that `(1 + sec theta - tan theta)/(1 + sec theta + tan theta) = (1 - sin theta)/cos theta`
Advertisements
उत्तर
L.H.S = `(1 + sec theta - tan theta)/(1 + sec theta + tan theta)`
= `(1 + 1//cos theta - sin theta//cos theta)/(1 + 1//cos theta + sin theta//cos theta)` ...`[∵ sec theta = 1/cos theta and tan theta = sin theta/cos theta]`
= `(cos theta + 1 - sin theta)/(cos theta + 1 + sin theta)`
= `((cos theta + 1) - sin theta)/((cos theta + 1) + sin theta)`
= `(2 cos^2 theta/2 - 2 sin theta/2 * cos theta/2)/(2 cos^2 theta/2 + 2 sin theta/2 * cos theta/2)` ...`[∵ 1 + cos theta = 2 cos^2 theta/2 and sin theta = 2sin theta/2 cos theta/2]`
= `(2cos^2 theta/2 - 2 sin theta/2 * cos theta/2)/(2cos^2 theta/2 + 2sin theta/2 * cos theta/2)`
= `(2cos theta/2 (cos theta/2 - sin theta/2))/(2cos theta/2(cos theta/2 + sin theta/2))`
= `(cos theta/2 - sin theta/2)/(cos theta/2 + sin theta/2) xx ((cos theta/2 - sin theta/2))/((cos theta/2 - sin theta/2))` ...[By rationalisation]
= `(cos theta/2 - sin theta/2)^2/((cos^2 theta/2 - sin^2 theta/2))` ...[∵ (a – b)2 = a2 + b2 – 2ab and (a – b)(a + b) = (a2 – b2)]
= `((cos^2 theta/2 + sin^2 theta/2) - (2 sin theta/2 * cos theta/2))/cos theta` ...`[∵ cos^2 theta/2 - sin^2 theta/2 = cos theta]`
= `(1 - sin theta)/cos theta` ...`[∵ sin^2 theta/2 + cos^2 theta/2 = 1]`
= R.H.S
Hence proved.
APPEARS IN
संबंधित प्रश्न
Evaluate
`(sin ^2 63^@ + sin^2 27^@)/(cos^2 17^@+cos^2 73^@)`
As observed from the top of an 80 m tall lighthouse, the angles of depression of two ships on the same side of the lighthouse of the horizontal line with its base are 30° and 40° respectively. Find the distance between the two ships. Give your answer correct to the nearest meter.
Prove the following trigonometric identities.
`(1 + sec theta)/sec theta = (sin^2 theta)/(1 - cos theta)`
Prove the following trigonometric identities.
`1/(sec A - 1) + 1/(sec A + 1) = 2 cosec A cot A`
Prove that: `sqrt((sec theta - 1)/(sec theta + 1)) + sqrt((sec theta + 1)/(sec theta - 1)) = 2 cosec theta`
Given that:
(1 + cos α) (1 + cos β) (1 + cos γ) = (1 − cos α) (1 − cos α) (1 − cos β) (1 − cos γ)
Show that one of the values of each member of this equality is sin α sin β sin γ
Prove the following identities:
`(cosecA - 1)/(cosecA + 1) = (cosA/(1 + sinA))^2`
If sin A + cos A = p and sec A + cosec A = q, then prove that : q(p2 – 1) = 2p.
Prove that:
(tan A + cot A) (cosec A – sin A) (sec A – cos A) = 1
`cot^2 theta - 1/(sin^2 theta ) = -1`a
`tan theta/(1+ tan^2 theta)^2 + cottheta/(1+ cot^2 theta)^2 = sin theta cos theta`
If `sec theta + tan theta = p,` prove that
(i)`sec theta = 1/2 ( p+1/p) (ii) tan theta = 1/2 ( p- 1/p) (iii) sin theta = (p^2 -1)/(p^2+1)`
Prove the following identity :
`sinA/(1 + cosA) + (1 + cosA)/sinA = 2cosecA`
If sinA + cosA = m and secA + cosecA = n , prove that n(m2 - 1) = 2m
If sec θ = `25/7`, then find the value of tan θ.
Prove that `tan^3 θ/( 1 + tan^2 θ) + cot^3 θ/(1 + cot^2 θ) = sec θ. cosec θ - 2 sin θ cos θ.`
Prove that `((1 + sin θ - cos θ)/( 1 + sin θ + cos θ))^2 = (1 - cos θ)/(1 + cos θ)`.
Prove that : `tan"A"/(1 - cot"A") + cot"A"/(1 - tan"A") = sec"A".cosec"A" + 1`.
Prove that `1/("cosec" theta - cot theta)` = cosec θ + cot θ
If tan θ + sec θ = l, then prove that sec θ = `(l^2 + 1)/(2l)`.
