Advertisements
Advertisements
Question
Prove the following:
`("cosec"theta + cottheta - 1)/( "cosec"theta + cot theta + 1) =(1-sintheta)/costheta`
Solution
We know that,
cot2θ = cosec2θ – 1
∴ cotθ · cotθ = (cosecθ + 1) (cosecθ –1)
∴ `cottheta/("cosec"theta + 1) = ("cosec"theta - 1)/cottheta`
By the theorem on equal ratios, we get
`cottheta/("cosec"theta + 1) = ("cosec"theta - 1) /cottheta = (cottheta + "cosec"theta - 1)/("cosec"theta +1 + cottheta)`
∴ `("cosec"theta- 1)/(cottheta) = (cottheta+"cosec"theta-1)/("cosec"theta + 1 + cot theta)`
∴ `(1/sintheta-1)/(costheta/sintheta)=("cosec"theta+cottheta-1)/("cosec"theta+cottheta+1)`
∴ `("cosec"theta + cottheta - 1)/( "cosec"theta + cot theta + 1) =(1-sintheta)/costheta`
APPEARS IN
RELATED QUESTIONS
If tanθ = `1/2`, evaluate `(2sin theta + 3cos theta)/(4cos theta + 3sin theta)`
Eliminate θ from the following:
x = 3secθ , y = 4tanθ
Eliminate θ from the following :
x = 4cosθ − 5sinθ, y = 4sinθ + 5cosθ
Eliminate θ from the following:
2x = 3 − 4 tan θ, 3y = 5 + 3 sec θ
Find the acute angle θ such that 2 cos2θ = 3 sin θ.
Find the acute angle θ such that 5tan2θ + 3 = 9secθ.
If cosecθ + cotθ = 5, then evaluate secθ.
Prove the following identities:
`(1 + tan^2 "A") + (1 + 1/tan^2"A") = 1/(sin^2 "A" - sin^4"A")`
Prove the following identities:
(sinθ + sec θ)2 + (cosθ + cosec θ)2 = (1 + cosecθ sec θ)2
Prove the following identities:
(1 + cot θ – cosec θ)(1 + tan θ + sec θ) = 2
Prove the following identities:
`tan^3theta/(1 + tan^2theta) + cot^3theta/(1 + cot^2theta` = secθ cosecθ – 2sinθ cosθ
Prove the following identities:
`sintheta/(1 + costheta) + (1 + costheta)/sintheta` = 2cosecθ
Prove the following identities:
`cottheta/("cosec" theta - 1) = ("cosec" theta + 1)/cot theta`
Prove the following identities:
(sec A + cos A)(sec A − cos A) = tan2A + sin2A
Prove the following identity:
1 + 3cosec2θ cot2θ + cot6θ = cosec6θ
Prove the following identities:
`(1 - sectheta + tan theta)/(1 + sec theta - tan theta) = (sectheta + tantheta - 1)/(sectheta + tantheta + 1)`
Select the correct option from the given alternatives:
If cosecθ + cotθ = `5/2`, then the value of tanθ is
Select the correct option from the given alternatives:
`1 - sin^2theta/(1 + costheta) + (1 + costheta)/sintheta - sintheta/(1 - costheta)` equals
Select the correct option from the given alternatives:
If cosecθ − cotθ = q, then the value of cot θ is
Prove the following:
2 sec2θ – sec4θ – 2cosec2θ + cosec4θ = cot4θ – tan4θ
Prove the following:
2(sin6θ + cos6θ) – 3(sin4θ + cos4θ) + 1 = 0
Prove the following:
tan2θ − sin2θ = sin4θ sec2θ
Prove the following:
(sinθ + cosecθ)2 + (cosθ + secθ)2 = tan2θ + cot2θ + 7
Prove the following:
sin8θ − cos8θ = (sin2θ − cos2θ) (1 − 2 sin2θ cos2θ)
Prove the following:
`(tantheta + sectheta - 1)/(tantheta + sectheta + 1) = tantheta/(sec theta + 1)`
Prove the following:
`("cosec"theta + cottheta + 1)/(cottheta + "cosec" theta - 1) = cottheta/("cosec"theta - 1)`
Prove the following identity:
`(1 - sec theta + tan theta)/(1 + sec theta - tan theta) = (sec theta + tan theta - 1)/(sec theta + tan theta + 1)`
If θ lies in the first quadrant and 5 tan θ = 4, then `(5 sin θ - 3 cos θ)/(sin θ + 2 cos θ)` is equal to ______.
