Advertisements
Advertisements
प्रश्न
Prove that `(1 + sin θ)/(1 - sin θ) = (sec θ + tan θ)^2`.
Advertisements
उत्तर
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
संबंधित प्रश्न
Prove that `\frac{\sin \theta -\cos \theta }{\sin \theta +\cos \theta }+\frac{\sin\theta +\cos \theta }{\sin \theta -\cos \theta }=\frac{2}{2\sin^{2}\theta -1}`
Prove the following trigonometric identities.
`"cosec" theta sqrt(1 - cos^2 theta) = 1`
Prove the following trigonometric identities.
`cos theta/(1 + sin theta) = (1 - sin theta)/cos theta`
If 3 sin θ + 5 cos θ = 5, prove that 5 sin θ – 3 cos θ = ± 3.
If cos θ + cot θ = m and cosec θ – cot θ = n, prove that mn = 1
Prove that:
`(cos^3A + sin^3A)/(cosA + sinA) + (cos^3A - sin^3A)/(cosA - sinA) = 2`
If x = a cos θ and y = b cot θ, show that:
`a^2/x^2 - b^2/y^2 = 1`
Show that none of the following is an identity:
`cos^2theta + cos theta = 1`
If `cos theta = 2/3 , " write the value of" (4+4 tan^2 theta).`
From the figure find the value of sinθ.
If sin2 θ cos2 θ (1 + tan2 θ) (1 + cot2 θ) = λ, then find the value of λ.
\[\frac{\tan \theta}{\sec \theta - 1} + \frac{\tan \theta}{\sec \theta + 1}\] is equal to
If tan θ = 2, where θ is an acute angle, find the value of cos θ.
Prove that tan2Φ + cot2Φ + 2 = sec2Φ.cosec2Φ.
Prove that `(tan θ)/(cot(90° - θ)) + (sec (90° - θ) sin (90° - θ))/(cosθ. cosec θ) = 2`.
If x = h + a cos θ, y = k + b sin θ.
Prove that `((x - h)/a)^2 + ((y - k)/b)^2 = 1`.
Prove the following identities: cot θ - tan θ = `(2 cos^2 θ - 1)/(sin θ cos θ)`.
sin2θ + sin2(90 – θ) = ?
Prove that `sqrt((1 + cos A)/(1 - cos A)) = "cosec" A + cot A`.
Prove that `(cot A)/(1 - tan A) + (tan A)/(1 - cot A) = 1 + tan A + cot A = sec A . "cosec" A + 1`.
