Advertisements
Advertisements
प्रश्न
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`
उत्तर
Given:
`x = a sec theta cos phi`
`=> x/a = sec theta cos phi` ........(1)
`y = b sec theta sin phi`
`=> y/b = sec theta sin phi`
`=> y/b = sec theta sin phi`
`=> zx/c = tan theta`
We have to prove that `x^2/a^2 + y^2/b^2 - z^2/c^2 = 1`
Squaring the above equations and then subtracting the third from the sum of the first two, we have
`(x/a)^2 + (y/b)^2 - (z/c)^2 = (sec theta cos phi)^2 + (sec theta sin phi)^2 - (tan theta)^2`
`=> x^2/ a^2 + y^2/b^2 - z^2/c62 = sec^2 theta cos^2 phi + sec^2 theta sin^2 phi - tan^2 theta`
`=> x^2/a^2 + y^2/b^2 - z^2/c^2 = (sec^2 theta cos^2 phi + sec^2 theta sin&2 phi) - tan^2 theta`
`=> x^2/a^2 + y^2/b^2 - z^2/c^2 = sec^2 theta(cos^2 phi + sin^2 phi) - tan^2 theta`
`=> x^2/a^2 + y^2/b^2 - z^2/c^2= sec^2 theta (1) = tan^2 theta`
`=> x^2/a^2 + y^2/b^2 - z^2/c^2 = sec^2 theta - tan^2 theta`
`=> x^2/a^2 + y^2/b^2 - z^2/c^2 = 1`
Hence proved.
APPEARS IN
संबंधित प्रश्न
Prove the following identities, where the angles involved are acute angles for which the expressions are defined:
`(sin theta-2sin^3theta)/(2cos^3theta -costheta) = tan theta`
Prove the identity (sin θ + cos θ)(tan θ + cot θ) = sec θ + cosec θ.
Without using trigonometric tables evaluate
`(sin 35^@ cos 55^@ + cos 35^@ sin 55^@)/(cosec^2 10^@ - tan^2 80^@)`
Prove the following trigonometric identities.
`(cos^2 theta)/sin theta - cosec theta + sin theta = 0`
Prove the following trigonometric identities
tan2 A + cot2 A = sec2 A cosec2 A − 2
Prove the following identities:
cosec A(1 + cos A) (cosec A – cot A) = 1
Prove the following identities:
`cosecA - cotA = sinA/(1 + cosA)`
`(tan^2theta)/((1+ tan^2 theta))+ cot^2 theta/((1+ cot^2 theta))=1`
`(cos theta cosec theta - sin theta sec theta )/(costheta + sin theta) = cosec theta - sec theta`
Write the value of `3 cot^2 theta - 3 cosec^2 theta.`
If sec2 θ (1 + sin θ) (1 − sin θ) = k, then find the value of k.
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 the following identity :
tanA+cotA=secAcosecA
Prove that `(tan θ)/(cot(90° - θ)) + (sec (90° - θ) sin (90° - θ))/(cosθ. cosec θ) = 2`.
If A = 60°, B = 30° verify that tan( A - B) = `(tan A - tan B)/(1 + tan A. tan B)`.
Prove that: `(1 + cot^2 θ/(1 + cosec θ)) = cosec θ`.
Prove the following identities.
`sqrt((1 + sin theta)/(1 - sin theta)) + sqrt((1 - sin theta)/(1 + sin theta))` = 2 sec θ
Prove that `costheta/(1 + sintheta) = (1 - sintheta)/(costheta)`
Prove that `(1 + sec "A")/"sec A" = (sin^2"A")/(1 - cos"A")`
If 1 + sin2θ = 3sinθ cosθ, then prove that tanθ = 1 or `1/2`.
