Advertisements
Advertisements
प्रश्न
Prove that `sqrt((1 - sin θ)/(1 + sin θ)) = sec θ - tan θ`.
Advertisements
उत्तर १
L.H.S. = `sqrt(((1 - sin θ)(1 - sin θ))/((1 + sin θ)(1 - sin θ)))`
= `sqrt((1 + sin^2θ - 2sinθ)/(1 - sin^2θ)`
= `sqrt((1 + sin^2θ - 2sinθ)/(cos^2θ)`
= `sqrt( 1/cos^2θ + sin^2θ/cos^2θ - (2sin θ)/cos θ xx 1/cosθ`
= `sqrt( sec^2θ + tan^2 θ - 2 tan θ . sec θ)`
= `sqrt((sec θ - tan θ)^2)`
= sec θ – tan θ
= R.H.S.
Hence proved.
उत्तर २
L.H.S. = `sqrt((1 - sin θ)/(1 + sin θ))`
= `sqrt(((1 - sin θ)(1 - sin θ))/((1 + sin θ)(1 - sin θ))`
= `sqrt(((1 - sin θ)^2)/(1 - sin^2θ)`
= `sqrt(((1 - sin θ)^2)/(cos^2θ)`
= `(1 - sin θ)/(cos θ)`
= `1/(cos θ) - (sin θ)/(cos θ)`
= sec θ – tan θ
= R.H.S.
Hence Proved.
APPEARS IN
संबंधित प्रश्न
If sinθ + cosθ = p and secθ + cosecθ = q, show that q(p2 – 1) = 2p
If secθ + tanθ = p, show that `(p^{2}-1)/(p^{2}+1)=\sin \theta`
Prove the following trigonometric identities.
`(tan^2 A)/(1 + tan^2 A) + (cot^2 A)/(1 + cot^2 A) = 1`
Prove the following trigonometric identities.
if `T_n = sin^n theta + cos^n theta`, prove that `(T_3 - T_5)/T_1 = (T_5 - T_7)/T_3`
Prove the following trigonometric identities.
`(cot^2 A(sec A - 1))/(1 + sin A) = sec^2 A ((1 - sin A)/(1 + sec A))`
Prove the following identities:
cosec A(1 + cos A) (cosec A – cot A) = 1
If x = r cos A cos B, y = r cos A sin B and z = r sin A, show that : x2 + y2 + z2 = r2
If `(cosec theta - sin theta )= a^3 and (sec theta - cos theta ) = b^3 , " prove that " a^2 b^2 ( a^2+ b^2 ) =1`
If \[\cos A = \frac{7}{25}\] find the value of tan A + cot A.
\[\frac{\tan \theta}{\sec \theta - 1} + \frac{\tan \theta}{\sec \theta + 1}\] is equal to
Prove the following Identities :
`(cosecA)/(cotA+tanA)=cosA`
Prove the following identity :
`(tanθ + 1/cosθ)^2 + (tanθ - 1/cosθ)^2 = 2((1 + sin^2θ)/(1 - sin^2θ))`
Without using trigonometric identity , show that :
`cos^2 25^circ + cos^2 65^circ = 1`
Prove that `(sin (90° - θ))/cos θ + (tan (90° - θ))/cot θ + (cosec (90° - θ))/sec θ = 3`.
Prove that `sin A/(sec A + tan A - 1) + cos A/(cosec A + cot A - 1) = 1`.
Prove the following identities.
`costheta/(1 + sintheta)` = sec θ – tan θ
Choose the correct alternative:
cos θ. sec θ = ?
If cos θ = `24/25`, then sin θ = ?
Prove that sec2θ − cos2θ = tan2θ + sin2θ
Let α, β be such that π < α – β < 3π. If sin α + sin β = `-21/65` and cos α + cos β = `-27/65`, then the value of `cos (α - β)/2` is ______.
