Advertisements
Advertisements
प्रश्न
Prove the following trigonometric identities.
(1 + tan2θ) (1 − sinθ) (1 + sinθ) = 1
Advertisements
उत्तर
We have to prove `(1 + tan^2 theta)(1 - sin theta)(1 + sin theta) = 1`
We know that
`sin^2 theta + cos^2 theta = 1`
`sec^2 theta - tan^2 theta = 1`
So
`(1 + tan^2 theta)(1 - sin theta) = (1 + tan^2 theta){(1 - sin theta)(1 + sin theta)}`
` = (1 + tan^2 theta)(1 - sin^2 theta)`
`= sec^2 theta cos^2 theta`
` = 1/cos^2 theta cos^2 theta`
= 1
APPEARS IN
संबंधित प्रश्न
Prove the following identities, where the angles involved are acute angles for which the expressions are defined:
`cos A/(1 + sin A) + (1 + sin A)/cos A = 2 sec A`
Prove the following trigonometric identities.
`(tan^2 A)/(1 + tan^2 A) + (cot^2 A)/(1 + cot^2 A) = 1`
If x = a sec θ cos ϕ, y = b sec θ sin ϕ and z = c tan θ, show that `x^2/a^2 + y^2/b^2 - x^2/c^2 = 1`
Prove the following identities:
`(cosecA - 1)/(cosecA + 1) = (cosA/(1 + sinA))^2`
Prove the following identities:
`sqrt((1 + sinA)/(1 - sinA)) = sec A + tan A`
If `cosA/cosB = m` and `cosA/sinB = n`, show that : (m2 + n2) cos2 B = n2.
`(sin theta +cos theta )/(sin theta - cos theta)+(sin theta- cos theta)/(sin theta + cos theta) = 2/((sin^2 theta - cos ^2 theta)) = 2/((2 sin^2 theta -1))`
Write the value of `(1 + cot^2 theta ) sin^2 theta`.
If `sin theta = x , " write the value of cot "theta .`
Prove that `(sinθ - cosθ + 1)/(sinθ + cosθ - 1) = 1/(secθ - tanθ)`
Prove the following identity :
`cosecA + cotA = 1/(cosecA - cotA)`
Prove that `(sec θ - 1)/(sec θ + 1) = ((sin θ)/(1 + cos θ ))^2`
Prove that: `(sec θ - tan θ)/(sec θ + tan θ ) = 1 - 2 sec θ.tan θ + 2 tan^2θ`
Prove that: sin6θ + cos6θ = 1 - 3sin2θ cos2θ.
Prove the following identities.
cot θ + tan θ = sec θ cosec θ
If cot θ + tan θ = x and sec θ – cos θ = y, then prove that `(x^2y)^(2/3) – (xy^2)^(2/3)` = 1
Choose the correct alternative:
cos θ. sec θ = ?
If tan θ = `9/40`, complete the activity to find the value of sec θ.
Activity:
sec2θ = 1 + `square` ......[Fundamental trigonometric identity]
sec2θ = 1 + `square^2`
sec2θ = 1 + `square`
sec θ = `square`
Prove that `sintheta/(sectheta+ 1) +sintheta/(sectheta - 1)` = 2 cot θ
If 4 tanβ = 3, then `(4sinbeta-3cosbeta)/(4sinbeta+3cosbeta)=` ______.
