Advertisements
Advertisements
Question
`Prove the following trigonometric identities.
`(sec A - tan A)^2 = (1 - sin A)/(1 + sin A)`
Advertisements
Solution
We need to prove `(sec A - tan A)^2 = (1 - sin A)/(1 + sin A)`
Here, we will first solve the L.H.S.
Now using `sec theta = 1/cos theta` and `tan theta = sin theta/cos theta` we get
`(sec A - tan A)^2 = (1/cos A - sin A/cos A)^2`
`= ((1 -sin A)/cos A)^2`
`= (1 - sin A)^2/(cos A)^2`
Further using the property `sin^2 theta + cos^2 theta = 1` we get
`((1 - sin A)^2/(cos A)) = (1 - sin A)^2/(1 - sin^2 A)`
`= (1 - sin A)^2/((1 - sin A)(1 + sin A))` (using `a^2 - b^2 = (a + b)(a - b))`
`= (1 - sin A)/(1 + sin A)`
henc e proved
APPEARS IN
RELATED QUESTIONS
Prove that ` \frac{\sin \theta -\cos \theta +1}{\sin\theta +\cos \theta -1}=\frac{1}{\sec \theta -\tan \theta }` using the identity sec2 θ = 1 + tan2 θ.
Prove the following trigonometric identities.
`cos theta/(1 + sin theta) = (1 - sin theta)/cos theta`
Prove the following trigonometric identities.
`sqrt((1 - cos A)/(1 + cos A)) = cosec A - cot A`
Prove the following trigonometric identities.
`(cos A cosec A - sin A sec A)/(cos A + sin A) = cosec A - sec A`
Prove the following trigonometric identities
If x = a sec θ + b tan θ and y = a tan θ + b sec θ, prove that x2 − y2 = a2 − b2
Prove the following identities:
cosec A(1 + cos A) (cosec A – cot A) = 1
Prove the following identities:
`((1 + tan^2A)cotA)/(cosec^2A) = tan A`
Prove the following identities:
(sin A + cosec A)2 + (cos A + sec A)2 = 7 + tan2 A + cot2 A
Prove the following identities:
`1/(1 - sinA) + 1/(1 + sinA) = 2sec^2A`
Prove the following identities:
cosec4 A (1 – cos4 A) – 2 cot2 A = 1
Prove that
`cot^2A-cot^2B=(cos^2A-cos^2B)/(sin^2Asin^2B)=cosec^2A-cosec^2B`
cos4 A − sin4 A is equal to ______.
(cosec θ − sin θ) (sec θ − cos θ) (tan θ + cot θ) is equal to
Prove the following identity :
cosecθ(1 + cosθ)(cosecθ - cotθ) = 1
prove that `1/(1 + cos(90^circ - A)) + 1/(1 - cos(90^circ - A)) = 2cosec^2(90^circ - A)`
If x = r sin θ cos Φ, y = r sin θ sin Φ and z = r cos θ, prove that x2 + y2 + z2 = r2.
Prove that sec θ. cosec (90° - θ) - tan θ. cot( 90° - θ ) = 1.
If 3 sin A + 5 cos A = 5, then show that 5 sin A – 3 cos A = ± 3
If cos A + cos2A = 1, then sin2A + sin4 A = ?
Complete the following activity to prove:
cotθ + tanθ = cosecθ × secθ
Activity: L.H.S. = cotθ + tanθ
= `cosθ/sinθ + square/cosθ`
= `(square + sin^2theta)/(sinθ xx cosθ)`
= `1/(sinθ xx cosθ)` ....... ∵ `square`
= `1/sinθ xx 1/cosθ`
= `square xx secθ`
∴ L.H.S. = R.H.S.
