Advertisements
Advertisements
प्रश्न
Prove the following identities:
`(1 + tan^2 "A") + (1 + 1/tan^2"A") = 1/(sin^2 "A" - sin^4"A")`
Advertisements
उत्तर
L.H.S. = `(1 + tan^2 "A") + (1 + 1/tan^2"A")`
= (sec2A) + (1 + cot2A)
= sec2A + cosec2A
= `1/cos^2"A" + 1/sin^2"A"`
= `(sin^2"A" + cos^2"A")/(sin^2"A" cos^2"A")`
= `1/(sin^2 "A" cos^2"A")`
= `1/(sin^2"A"(1 - sin^2"A")`
= `1/(sin^2"A" - sin^4"A")`
= R.H.S.
APPEARS IN
संबंधित प्रश्न
Evaluate the following:
sin 30° + cos 45° + tan 180°
Evaluate the following :
cosec 45° + cot 45° + tan 0°
Evaluate the following :
sin 30° × cos 45° × tan 360°
If tanθ = `1/2`, evaluate `(2sin theta + 3cos theta)/(4cos theta + 3sin theta)`
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:
(cos2A – 1) (cot2A + 1) = −1
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:
`tan"A"/(1 + sec"A") + (1 + sec"A")/tan"A"` is equal to
Select the correct option from the given alternatives:
If cosecθ + cotθ = `5/2`, then the value of tanθ is
Prove the following:
`((1 + cot theta + tan theta)(sin theta - costheta)) /(sec^3theta - "cosec"^3theta)`= sin2θ cos2θ
Prove the following:
2 sec2θ – sec4θ – 2cosec2θ + cosec4θ = cot4θ – tan4θ
Prove the following:
2(sin6θ + cos6θ) – 3(sin4θ + cos4θ) + 1 = 0
Prove the following:
cos4θ − sin4θ +1= 2cos2θ
Prove the following:
sin4θ +2sin2θ . cos2θ = 1 − cos4θ
Prove the following:
`(sin^3theta + cos^3theta)/(sintheta + costheta) + (sin^3theta - cos^3theta)/(sintheta - costheta)` = 2
Prove the following:
tan2θ − sin2θ = sin4θ sec2θ
Prove the following:
sin8θ − cos8θ = (sin2θ − cos2θ) (1 − 2 sin2θ cos2θ)
Prove the following:
sin6A + cos6A = 1 − 3sin2A + 3 sin4A
Prove the following:
`(1 + cottheta + "cosec" theta)/(1 - cottheta + "cosec" theta) = ("cosec" theta + cottheta - 1)/(cottheta - "cosec"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`
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 ______.
