Advertisements
Advertisements
Question
Prove that `(cosθ)/(1 + sinθ) = (1 - sinθ)/(cosθ)`.
Advertisements
Solution
L.H.S. = `(cosθ)/(1 + sinθ)`
= `(cosθ)/(1 + sinθ) xx (1 - sinθ)/(1 - sinθ)` ...[On rationalising the denominator]
= `(cosθ(1 - sinθ))/(1 - sin^2θ)`
= `(cosθ(1 - sinθ))/(cos^2θ)` ...`[(∵ sin^2θ + cos^2θ = 1),(∴ 1 -sin^2θ = cos^2θ)]`
= `(1 - sinθ)/(cosθ)`
= R.H.S.
∴ `(cosθ)/(1 + sinθ) = (1 - sinθ)/(cosθ)`
APPEARS IN
RELATED QUESTIONS
If cosθ + sinθ = √2 cosθ, show that cosθ – sinθ = √2 sinθ.
Prove the following trigonometric identity.
`cos^2 A + 1/(1 + cot^2 A) = 1`
if `x/a cos theta + y/b sin theta = 1` and `x/a sin theta - y/b cos theta = 1` prove that `x^2/a^2 + y^2/b^2 = 2`
Prove the following identities:
`(costhetacottheta)/(1 + sintheta) = cosectheta - 1`
Prove that:
`1/(cosA + sinA - 1) + 1/(cosA + sinA + 1) = cosecA + secA`
Prove the following identities:
`(1 + (secA - tanA)^2)/(cosecA(secA - tanA)) = 2tanA`
If tan A = n tan B and sin A = m sin B, prove that `cos^2A = (m^2 - 1)/(n^2 - 1)`
If 2 sin A – 1 = 0, show that: sin 3A = 3 sin A – 4 sin3 A
`(1+ cos theta - sin^2 theta )/(sin theta (1+ cos theta))= cot theta`
Write the value of tan10° tan 20° tan 70° tan 80° .
Prove the following identity:
`cosA/(1 + sinA) = secA - tanA`
Prove the following identity :
`(secA - 1)/(secA + 1) = (1 - cosA)/(1 + cosA)`
Find the value of x , if `cosx = cos60^circ cos30^circ - sin60^circ sin30^circ`
If cosθ = `5/13`, then find sinθ.
Prove that `(tan^2"A")/(tan^2 "A"-1) + (cosec^2"A")/(sec^2"A"-cosec^2"A") = (1)/(1-2 co^2 "A")`
Prove the following identities.
tan4 θ + tan2 θ = sec4 θ – sec2 θ
If sin θ + cos θ = `sqrt(3)`, then prove that tan θ + cot θ = 1.
Prove that `"cosec" θ xx sqrt(1 - cos^2θ) = 1`.
If cosec A – sin A = p and sec A – cos A = q, then prove that `(p^2q)^(2/3) + (pq^2)^(2/3) = 1`.
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`
