Advertisements
Advertisements
प्रश्न
If cosec θ + cot θ = p, then prove that cos θ = `(p^2 - 1)/(p^2 + 1)`
उत्तर
According to the question,
cosec θ + cot θ = p
Since, cosec θ = `1/sintheta` and cot θ = `costheta/sintheta`
`1/sintheta + costheta/sintheta` = p
`(1 + costheta)/sintheta` = p
Squaring on L.H.S and R.H.S,
`((1 + costheta)/sin theta)^2` = p2
`(1 + cos^2 theta + 2 cos theta)/(sin^2 theta)` = p2
Applying component and dividend rule,
`((1 + cos^2 theta + 2 cos theta) - sin^2 theta)/((1 + cos^2 theta + 2 cos theta) + sin^2 theta) = ("p"^2 - 1)/("p"^2 + 1)`
= `((1 - sin^2theta) + cos^2 theta + 2 cos theta)/(sin^2 theta + cos^2 theta + 1 + 2 cos theta) = ("p"^2 - 1)/("p"^2 + 1)`
Since, 1 – sin2θ = cos2θ and sin2θ + cos2θ = 1
`(cos^2 theta + cos^2 theta + 2 cos theta)/(1 + 1 + 2 cos theta) = ("p"^2 - 1)/("p"^2 + 1)`
`(2 cos^2 theta + 2 cos theta)/(2 + 2 cos theta) = ("p"^2 - 1)/("p"^2 + 1)`
`(2 cos theta(cos theta + 1))/(2(cos theta + 1)) = ("p"^2 - 1)/("p"^2 + 1)`
cos θ = `("p"^2 - 1)/("p"^2 + 1)`
Hence proved.
APPEARS IN
संबंधित प्रश्न
Prove the following trigonometric identity.
`(sin theta - cos theta + 1)/(sin theta + cos theta - 1) = 1/(sec theta - tan theta)`
`(sectheta- tan theta)/(sec theta + tan theta) = ( cos ^2 theta)/( (1+ sin theta)^2)`
Write the value of `(1 + tan^2 theta ) cos^2 theta`.
If x = a sin θ and y = bcos θ , write the value of`(b^2 x^2 + a^2 y^2)`
What is the value of 9cot2 θ − 9cosec2 θ?
\[\frac{x^2 - 1}{2x}\] is equal to
9 sec2 A − 9 tan2 A is equal to
\[\frac{1 + \tan^2 A}{1 + \cot^2 A}\]is equal to
Prove the following identity :
`sinA/(1 + cosA) + (1 + cosA)/sinA = 2cosecA`
Prove that:
`sqrt((sectheta - 1)/(sec theta + 1)) + sqrt((sectheta + 1)/(sectheta - 1)) = 2cosectheta`
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 `cos θ/sin(90° - θ) + sin θ/cos (90° - θ) = 2`.
Prove that: `1/(sec θ - tan θ) = sec θ + tan θ`.
Prove the following identities.
`sqrt((1 + sin theta)/(1 - sin theta)) + sqrt((1 - sin theta)/(1 + sin theta))` = 2 sec θ
Choose the correct alternative:
tan (90 – θ) = ?
Choose the correct alternative:
Which is not correct formula?
The value of tan A + sin A = M and tan A - sin A = N.
The value of `("M"^2 - "N"^2) /("MN")^0.5`
Given that sinθ + 2cosθ = 1, then prove that 2sinθ – cosθ = 2.
(tan θ + 2)(2 tan θ + 1) = 5 tan θ + sec2θ.
If sin θ + cos θ = p and sec θ + cosec θ = q, then prove that q(p2 – 1) = 2p.
