Advertisements
Advertisements
प्रश्न
Prove the following identities:
(sinθ + sec θ)2 + (cosθ + cosec θ)2 = (1 + cosecθ sec θ)2
उत्तर
L.H.S. = (sinθ + sec θ)2 + (cosθ + cosec θ)2
= `(sin theta + 1/cos theta)^2 + (cos theta + 1/sin theta)^2`
= `(sin theta cos theta + 1)^2/cos^2 theta + (sin theta cos theta + 1)^2/sin^2 theta `
= `(sin theta cos theta + 1)^2 (1/cos^2theta + 1/sin^2 theta)`
= `(sin theta cos theta + 1)^2 ((sin^2 theta + cos^2 theta)/(sin^2 theta cos^2 theta))`
= `(sin theta cos theta + 1)^2 (1/(sin^2 theta cos^2 theta))`
= `((sin theta cos theta + 1)/(sin theta cos theta))^2`
= `((sin theta cos theta)/(sin theta cos theta) + 1/(sin theta cos theta))^2`
= (1 + cosecθ secθ)2
= R.H.S.
APPEARS IN
संबंधित प्रश्न
Eliminate θ from the following :
x = 4cosθ − 5sinθ, y = 4sinθ + 5cosθ
Eliminate θ from the following :
x = 5 + 6cosecθ, y = 3 + 8cotθ
Eliminate θ from the following:
2x = 3 − 4 tan θ, 3y = 5 + 3 sec θ
Find sinθ such that 3cosθ + 4sinθ = 4
If cosecθ + cotθ = 5, then evaluate secθ.
If cotθ = `3/4` and π < θ < `(3pi)/2` then find the value of 4cosecθ + 5cosθ.
Prove the following identities:
(cos2A – 1) (cot2A + 1) = −1
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
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
Select the correct option from the given alternatives:
The value of tan1°.tan2°tan3°..... tan89° is equal to
Prove the following:
`((1 + cot theta + tan theta)(sin theta - costheta)) /(sec^3theta - "cosec"^3theta)`= sin2θ cos2θ
Prove the following:
sin4θ + cos4θ = 1 – 2 sin2θ cos2θ
Prove the following:
2(sin6θ + cos6θ) – 3(sin4θ + cos4θ) + 1 = 0
Prove the following:
cos4θ − sin4θ +1= 2cos2θ
Prove the following:
`(sin^3theta + cos^3theta)/(sintheta + costheta) + (sin^3theta - cos^3theta)/(sintheta - costheta)` = 2
Prove the following:
sin6A + cos6A = 1 − 3sin2A + 3 sin4A
Prove the following:
(1 + tanA · tanB)2 + (tanA − tanB)2 = sec2A · sec2B
Prove the following:
`(1 + cot + "cosec" theta)/(1 - cot + "cosec" theta) = ("cosec" theta + cottheta - 1)/(cottheta - "cosec"theta + 1)`
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:
`("cosec"theta + cottheta - 1)/( "cosec"theta + cot theta + 1) =(1-sintheta)/costheta`
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 ______.
