Advertisements
Advertisements
प्रश्न
Prove the following trigonometric identities.
`(1 + cos theta - sin^2 theta)/(sin theta (1 + cos theta)) = cot theta`
Advertisements
उत्तर
In the given question, we need to prove `(1 + cos theta - sin^2 theta)/(sin theta (1 + cos theta)) = cot theta`
Using the property `sin^2 theta + cot^2 theta = 1` we get
So
`(1 + cos theta - sin^2 theta)/(sin theta (1 + cos theta))`
`= (1 + cos theta - (1 - cos^2 theta))/(sin theta (1 + cos theta)`
`= (cos theta + cos^2 theta)/(sin theta (1 + cos theta))`
Solving further, we get
`(cos theta + cos^2 theta)/(sin(1 + cos theta)) = (cos theta (1 + cos theta))/(sin theta(1 + cos theta))`
`= cos theta/sin theta`
`= cot theta`
Hence proved.
APPEARS IN
संबंधित प्रश्न
The angles of depression of two ships A and B as observed from the top of a light house 60 m high are 60° and 45° respectively. If the two ships are on the opposite sides of the light house, find the distance between the two ships. Give your answer correct to the nearest whole number.
As observed from the top of an 80 m tall lighthouse, the angles of depression of two ships on the same side of the lighthouse of the horizontal line with its base are 30° and 40° respectively. Find the distance between the two ships. Give your answer correct to the nearest meter.
Prove the following trigonometric identities.
`"cosec" theta sqrt(1 - cos^2 theta) = 1`
Prove the following trigonometric identities.
`(1 - tan^2 A)/(cot^2 A -1) = tan^2 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
If a cos θ + b sin θ = m and a sin θ – b cos θ = n, prove that a2 + b2 = m2 + n2
Show that : tan 10° tan 15° tan 75° tan 80° = 1
`sec theta (1- sin theta )( sec theta + tan theta )=1`
If `( cos theta + sin theta) = sqrt(2) sin theta , " prove that " ( sin theta - cos theta ) = sqrt(2) cos theta`
(cosec θ − sin θ) (sec θ − cos θ) (tan θ + cot θ) is equal to
Prove the following identity :
`1/(tanA + cotA) = sinAcosA`
Prove the following identity :
`sec^4A - sec^2A = sin^2A/cos^4A`
Without using trigonometric table , evaluate :
`cos90^circ + sin30^circ tan45^circ cos^2 45^circ`
Prove that: 2(sin6θ + cos6θ) - 3 ( sin4θ + cos4θ) + 1 = 0.
1 + cot2θ = ?
If 5 sec θ – 12 cosec θ = 0, then find values of sin θ, sec θ.
Prove that `(1 + sin B)/(cos B) + (cos B)/(1 + sin B) = 2 sec B`.
Prove that (1 – cos2A) . sec2B + tan2B (1 – sin2A) = sin2A + tan2B.
If tan θ = 3, then `(4 sin theta - cos theta)/(4 sin theta + cos theta)` is equal to ______.
If cot θ = `40/9`, find the values of cosec θ and sinθ,
We have, 1 + cot2θ = cosec2θ
1 + `square` = cosec2θ
1 + `square` = cosec2θ
`(square + square)/square` = cosec2θ
`square/square` = cosec2θ ......[Taking root on the both side]
cosec θ = `41/9`
and sin θ = `1/("cosec" θ)`
sin θ = `1/square`
∴ sin θ = `9/41`
The value is cosec θ = `41/9`, and sin θ = `9/41`
