Advertisements
Advertisements
प्रश्न
`Prove the following trigonometric identities.
`(sec A - tan A)^2 = (1 - sin A)/(1 + sin A)`
Advertisements
उत्तर
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
संबंधित प्रश्न
If cosθ + sinθ = √2 cosθ, show that cosθ – sinθ = √2 sinθ.
Prove the following identities, where the angles involved are acute angles for which the expressions are defined:
`(cos A-sinA+1)/(cosA+sinA-1)=cosecA+cotA ` using the identity cosec2 A = 1 cot2 A.
Prove that `sqrt(sec^2 theta + cosec^2 theta) = tan theta + cot theta`
Prove the following trigonometric identities.
sec A (1 − sin A) (sec A + tan A) = 1
Prove the following identities:
`(sinAtanA)/(1 - cosA) = 1 + secA`
Prove the following identities:
`1 - cos^2A/(1 + sinA) = sinA`
Prove that:
`1/(cosA + sinA - 1) + 1/(cosA + sinA + 1) = cosecA + secA`
Prove the following identities:
`cosA/(1 + sinA) + tanA = secA`
Prove that:
`cosA/(1 + sinA) = secA - tanA`
`cot^2 theta - 1/(sin^2 theta ) = -1`a
Four alternative answers for the following question are given. Choose the correct alternative and write its alphabet:
sin θ × cosec θ = ______
If a cos θ + b sin θ = 4 and a sin θ − b sin θ = 3, then a2 + b2 =
If sin θ + sin2 θ = 1, then cos2 θ + cos4 θ =
Prove the following identity :
`(1 + tan^2A) + (1 + 1/tan^2A) = 1/(sin^2A - sin^4A)`
Prove that `(tan^2 theta - 1)/(tan^2 theta + 1)` = 1 – 2 cos2θ
Choose the correct alternative:
`(1 + cot^2"A")/(1 + tan^2"A")` = ?
Prove that `1/("cosec" theta - cot theta)` = cosec θ + cot θ
Given that sin θ = `a/b`, then cos θ is equal to ______.
Prove the following:
`sintheta/(1 + cos theta) + (1 + cos theta)/sintheta` = 2cosecθ
Show that, cotθ + tanθ = cosecθ × secθ
Solution :
L.H.S. = cotθ + tanθ
= `cosθ/sinθ + sinθ/cosθ`
= `(square + square)/(sinθ xx cosθ)`
= `1/(sinθ xx cosθ)` ............... `square`
= `1/sinθ xx 1/square`
= cosecθ × secθ
L.H.S. = R.H.S
∴ cotθ + tanθ = cosecθ × secθ
