Advertisements
Advertisements
प्रश्न
Prove the following:
`tanA/(1 + sec A) - tanA/(1 - sec A)` = 2cosec A
Advertisements
उत्तर
L.H.S:
`tanA/(1 + sec A) - tanA/(1 - sec A)`
Taking LCM of the denominators,
= `(tanA(1 - sec A) - tanA(1 + sec A))/((1 + sec A)(1 - sec A))`
Since, (1 + sec A)(1 – sec A) = 1 – sec2A
= `(tan A(1 - secA - 1 - sec A))/(1 - sec^2A)`
= `(tan A(-2 sec A))/(1 - sec^2 A)`
= `(2 tan A *sec A)/(sec^2 A - 1)`
Since,
sec2A – tan2A = 1
sec2A – 1 = tan2A
= `(2 tan A * sec A)/(tan^2 A)`
Since, sec A = `(1/cosA)` and tan A = `(sinA/cosA)`
= `(2secA)/tanA = (2cosA)/(cosA sinA)`
= `2/sinA`
= 2 cosec A ...`(∵ 1/sinA = "cosec" A)`
= R.H.S
Hence proved.
APPEARS IN
संबंधित प्रश्न
Prove the following trigonometric identities
(1 + cot2 A) sin2 A = 1
Prove the following trigonometric identities.
`1/(sec A - 1) + 1/(sec A + 1) = 2 cosec A cot A`
Prove the following identities:
(1 + cot A – cosec A)(1 + tan A + sec A) = 2
Prove the following identities:
`sinA/(1 - cosA) - cotA = cosecA`
Prove the following identities:
`(1 - 2sin^2A)^2/(cos^4A - sin^4A) = 2cos^2A - 1`
`costheta/((1-tan theta))+sin^2theta/((cos theta-sintheta))=(cos theta+ sin theta)`
If `tan theta = 1/sqrt(5), "write the value of" (( cosec^2 theta - sec^2 theta))/(( cosec^2 theta - sec^2 theta))`.
Find the value of `(cos 38° cosec 52°)/(tan 18° tan 35° tan 60° tan 72° tan 55°)`
Prove that `(sinθ - cosθ + 1)/(sinθ + cosθ - 1) = 1/(secθ - tanθ)`
If a cot θ + b cosec θ = p and b cot θ − a cosec θ = q, then p2 − q2
If x = a sec θ cos ϕ, y = b sec θ sin ϕ and z = c tan θ, then\[\frac{x^2}{a^2} + \frac{y^2}{b^2}\]
Prove that:
`(cot A - 1)/(2 - sec^2 A) = cot A/(1 + tan A)`
If A + B = 90°, show that sec2 A + sec2 B = sec2 A. sec2 B.
Prove the following identities:
`1/(sin θ + cos θ) + 1/(sin θ - cos θ) = (2sin θ)/(1 - 2 cos^2 θ)`.
Prove that `[(1 + sin theta - cos theta)/(1 + sin theta + cos theta)]^2 = (1 - cos theta)/(1 + cos theta)`
If tan θ = `13/12`, then cot θ = ?
Prove that `"cosec" θ xx sqrt(1 - cos^2theta)` = 1
sin(45° + θ) – cos(45° – θ) is equal to ______.
Prove that `(1 + tan^2 A)/(1 + cot^2 A)` = sec2 A – 1
