Advertisements
Advertisements
प्रश्न
Prove the following identities:
`1/(sectheta + tantheta) - 1/costheta = 1/costheta - 1/(sectheta - tantheta)`
Advertisements
उत्तर
We have to prove that,
`1/(sectheta + tantheta) - 1/costheta = 1/costheta - 1/(sectheta - tantheta)`
`1/(sectheta + tantheta) + 1/(sectheta - tantheta) = 1/costheta + 1/costheta`
`1/(sectheta + tantheta) + 1/(sec theta - tan theta) = 2/costheta`
L.H.S. = `1/(sectheta + tantheta) + 1/(sec theta - tan theta)`
= `(sectheta - tantheta + sectheta + tantheta)/((sectheta + tantheta)(sectheta - tantheta))`
= `(2sectheta)/(sec^2theta - tan^2theta)` .....[a2 − b2 = (a + b) (a − b)]
= `(2secθ)/1` ...`[(because sec^2theta = 1 + tan^2theta),(therefore sec^2 theta - tan^2theta = 1)]`
= 2secθ
= `2 xx 1/(cosθ)`
= `2/costheta`
= R.H.S.
Hence proved.
APPEARS IN
संबंधित प्रश्न
Evaluate the following:
sin 30° + cos 45° + tan 180°
If tanθ = `1/2`, evaluate `(2sin theta + 3cos theta)/(4cos theta + 3sin theta)`
Eliminate θ from the following :
x = 4cosθ − 5sinθ, y = 4sinθ + 5cosθ
Eliminate θ from the following :
x = 5 + 6cosecθ, y = 3 + 8cotθ
Find the acute angle θ such that 2 cos2θ = 3 sin θ.
If cotθ = `3/4` and π < θ < `(3pi)/2` then find the value of 4cosecθ + 5cosθ.
Prove the following identities:
`(1 + tan^2 "A") + (1 + 1/tan^2"A") = 1/(sin^2 "A" - sin^4"A")`
Prove the following identities:
(cos2A – 1) (cot2A + 1) = −1
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:
`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 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:
`1 - sin^2theta/(1 + costheta) + (1 + costheta)/sintheta - sintheta/(1 - costheta)` equals
Prove the following:
`((1 + cot theta + tan theta)(sin theta - costheta)) /(sec^3theta - "cosec"^3theta)`= sin2θ cos2θ
Prove the following:
`(tan theta + 1/costheta)^2 + (tan theta - 1/costheta)^2 = 2((1 + sin^2theta)/(1 - sin^2theta))`
Prove the following:
2 sec2θ – sec4θ – 2cosec2θ + cosec4θ = cot4θ – tan4θ
Prove the following:
sin4θ + cos4θ = 1 – 2 sin2θ cos2θ
Prove the following:
2(sin6θ + cos6θ) – 3(sin4θ + cos4θ) + 1 = 0
Prove the following:
sin4θ +2sin2θ . cos2θ = 1 − cos4θ
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:
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 identity:
`(1 - sec theta + tan theta)/(1 + sec theta - tan theta) = (sec theta + tan theta - 1)/(sec theta + tan theta + 1)`
If 5 tan θ = 4. then `(5 sin θ − 3 cos θ)/(5 sin θ + 2 cos θ)` = ______.
