Advertisements
Advertisements
प्रश्न
Prove the following trigonometric identities.
`(1 + cot A + tan A)(sin A - cos A) = sec A/(cosec^2 A) - (cosec A)/sec^2 A = sin A tan A - cos A cot A`
उत्तर
We have prove that
`(1 + cot A + tan A)(sin A - cos A) = sec A/(cosec^2 A) - (cosec A)/sec^2 A = sin A tan A - cos A cot A`
We know that `sin^2 A + cos^2 A = 1`
So,
(2 + cot A + tan A)(sin A - cos A)
`= (1 + cos A/sin A + sin A/cos A)(sin A - cos A)`
`= ((sin A cos A + cos^2 A + sin^2 A)/(sin A cos A)) (sin A - cos A)`
`= ((sin A cos A + 1)/(sin A cos A))(sin A - cos A)`
`= ((sin A - cos A)(sin A cos A + 1))/(sin A cos A)`
`= (sin^2 A cos A + sin A - cos^2 A sin A - cos A)/(sin A cos A)`
`= ((sin^2 A cos A - cos A) + (sin A - cos^2 A sin A))/(sin A cos A)`
`= (cos A (sin^2 A - 1)+ sin A (1 - sin^2 A))/(sin A cos A)`
`= (cos A (-cos^2 A) + sin A (sin^2 A))/(sin A cos A)`
`= (-cos^3 A + sin^3 A)/(sin A cos A)`
`= (sin^3 A - cos^3 A)/(sin A cos A)`
`= (sin^2 A)/cos A - cos^2 A/sin A`
`= sin A/cos A sin A - cos A/sin A cos A`
`= tan A sin A - cot A cos A`
= sin A tan A - cos A cot A
Now
`sec A/(cosec^2 A) - (cosec A)/sec^2 A = (1/cos A)/(1/sin^2 A) - (1/sin A)/(1/cos^2 A)`
`= sin^2 A/cos A - cos^2 A/sin A`
`= sin A sin A/cos a - cos A cos A/sin A`
= sin A tan A - cos A cot A
Hence proved.
APPEARS IN
संबंधित प्रश्न
Prove that (1 + cot θ – cosec θ)(1+ tan θ + sec θ) = 2
Prove the following trigonometric identities
`(1 + tan^2 theta)/(1 + cot^2 theta) = ((1 - tan theta)/(1 - cot theta))^2 = tan^2 theta`
Prove the following trigonometric identities.
`sqrt((1 - cos A)/(1 + cos A)) = cosec A - cot A`
Prove the following trigonometric identities.
`(1 - tan^2 A)/(cot^2 A -1) = tan^2 A`
If 3 sin θ + 5 cos θ = 5, prove that 5 sin θ – 3 cos θ = ± 3.
If x= a sec `theta + b tan theta and y = a tan theta + b sec theta ,"prove that" (x^2 - y^2 )=(a^2 -b^2)`
Write True' or False' and justify your answer the following :
The value of sin θ+cos θ is always greater than 1 .
Prove the following identity :
`(1 + sinA)/(1 - sinA) = (cosecA + 1)/(cosecA - 1)`
There are two poles, one each on either bank of a river just opposite to each other. One pole is 60 m high. From the top of this pole, the angle of depression of the top and foot of the other pole are 30° and 60° respectively. Find the width of the river and height of the other pole.
Prove that tan2Φ + cot2Φ + 2 = sec2Φ.cosec2Φ.
`(sin A)/(1 + cos A) + (1 + cos A)/(sin A)` = 2 cosec A
Prove that `( 1 + sin θ)/(1 - sin θ) = 1 + 2 tan θ/cos θ + 2 tan^2 θ` .
If tan A + sin A = m and tan A - sin A = n, then show that m2 - n2 = 4 `sqrt(mn)`.
Prove that `((1 - cos^2 θ)/cos θ)((1 - sin^2θ)/(sin θ)) = 1/(tan θ + cot θ)`
Without using a trigonometric table, prove that
`(cos 70°)/(sin 20°) + (cos 59°)/(sin 31°) - 8sin^2 30° = 0`.
If sin θ (1 + sin2 θ) = cos2 θ, then prove that cos6 θ – 4 cos4 θ + 8 cos2 θ = 4
Prove that `"cot A"/(1 - cot"A") + "tan A"/(1 - tan "A")` = – 1
If tan θ = 3, then `(4 sin theta - cos theta)/(4 sin theta + cos theta)` is equal to ______.
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.
If sinθ = `11/61`, then find the value of cosθ using the trigonometric identity.
