Advertisements
Advertisements
प्रश्न
If cosec θ + cot θ = p, then prove that cos θ = `(p^2 - 1)/(p^2 + 1)`
Advertisements
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 that sin6θ + cos6θ = 1 – 3 sin2θ. cos2θ.
Prove the following identities:
(cosec A – sin A) (sec A – cos A) (tan A + cot A) = 1
Prove the following identities:
`1/(secA + tanA) = secA - tanA`
If m = a sec A + b tan A and n = a tan A + b sec A, then prove that : m2 – n2 = a2 – b2
Show that : `sinAcosA - (sinAcos(90^circ - A)cosA)/sec(90^circ - A) - (cosAsin(90^circ - A)sinA)/(cosec(90^circ - A)) = 0`
`cot^2 theta - 1/(sin^2 theta ) = -1`a
If m = ` ( cos theta - sin theta ) and n = ( cos theta + sin theta ) "then show that" sqrt(m/n) + sqrt(n/m) = 2/sqrt(1-tan^2 theta)`.
If sec2 θ (1 + sin θ) (1 − sin θ) = k, then find the value of k.
Write True' or False' and justify your answer the following :
The value of sin θ+cos θ is always greater than 1 .
If x = a cos θ and y = b sin θ, then b2x2 + a2y2 =
Prove the following identity :
`(1 + tan^2A) + (1 + 1/tan^2A) = 1/(sin^2A - sin^4A)`
Prove that `sin(90^circ - A).cos(90^circ - A) = tanA/(1 + tan^2A)`
For ΔABC , prove that :
`tan ((B + C)/2) = cot "A/2`
If A = 30°, verify that `sin 2A = (2 tan A)/(1 + tan^2 A)`.
Prove that: `(sin θ - 2sin^3 θ)/(2 cos^3 θ - cos θ) = tan θ`.
Prove that: sin6θ + cos6θ = 1 - 3sin2θ cos2θ.
a cot θ + b cosec θ = p and b cot θ + a cosec θ = q then p2 – q2 is equal to
Choose the correct alternative:
1 + cot2θ = ?
Choose the correct alternative:
tan (90 – θ) = ?
If tan θ = `x/y`, then cos θ is equal to ______.
