Advertisements
Advertisements
प्रश्न
Prove the following identities, where the angles involved are acute angles for which the expressions are defined:
`(tan theta)/(1-cot theta) + (cot theta)/(1-tan theta) = 1+secthetacosectheta`
[Hint: Write the expression in terms of sinθ and cosθ]
Advertisements
उत्तर
L.H.S
= `(tantheta)/(1-cottheta) + (cottheta)/(1-tantheta) `
= `(sintheta/costheta)/(1-costheta/sintheta) + (costheta/sintheta)/(1-sintheta/costheta)`
= `(sintheta/costheta)/((sintheta-costheta)/(sintheta))+ (costheta/sintheta)/((costheta-sintheta)/costheta)`
= `(sin^2theta)/(costheta(sintheta-costheta)) - (cos^2theta)/(sintheta(sintheta-costheta))`
= `1/(sintheta - costheta)[(sin^2theta)/costheta - cos^2theta/sintheta]`
= `(1/(sintheta-costheta))[(sin^3theta-cos^3theta)/(sinthetacostheta)]`
= `(1/(sintheta-costheta))[((sintheta-costheta)(sin^2theta+cos^2theta+sinthetacostheta))/(sinthetacostheta)]`
= `((1+sinthetacostheta))/((sinthetacostheta))`
= sec θ cosec θ + 1
= R.H.S
APPEARS IN
संबंधित प्रश्न
Prove the following identities, where the angles involved are acute angles for which the expressions are defined:
`(cosec θ – cot θ)^2 = (1-cos theta)/(1 + cos theta)`
Prove the following trigonometric identities.
`tan theta/(1 - cot theta) + cot theta/(1 - tan theta) = 1 + tan theta + cot theta`
Prove the following trigonometric identities.
sec6θ = tan6θ + 3 tan2θ sec2θ + 1
Prove the following identities:
`1/(1 + cosA) + 1/(1 - cosA) = 2cosec^2A`
`(cot ^theta)/((cosec theta+1)) + ((cosec theta + 1))/cot theta = 2 sec theta`
`(cos^3 theta +sin^3 theta)/(cos theta + sin theta) + (cos ^3 theta - sin^3 theta)/(cos theta - sin theta) = 2`
` (sin theta + cos theta )/(sin theta - cos theta ) + ( sin theta - cos theta )/( sin theta + cos theta) = 2/ ((1- 2 cos^2 theta))`
`(sin theta)/((sec theta + tan theta -1)) + cos theta/((cosec theta + cot theta -1))=1`
Write the value of `(1 + tan^2 theta ) cos^2 theta`.
Prove the following identity :
`(1 - sin^2θ)sec^2θ = 1`
Prove the following identity :
`(cot^2θ(secθ - 1))/((1 + sinθ)) = sec^2θ((1-sinθ)/(1 + secθ))`
Prove that sec θ. cosec (90° - θ) - tan θ. cot( 90° - θ ) = 1.
Prove the following identities.
cot θ + tan θ = sec θ cosec θ
Choose the correct alternative:
tan (90 – θ) = ?
Prove that `(cos(90 - "A"))/(sin "A") = (sin(90 - "A"))/(cos "A")`
Prove that `sintheta/(sectheta+ 1) +sintheta/(sectheta - 1)` = 2 cot θ
If tan θ = 3, then `(4 sin theta - cos theta)/(4 sin theta + cos theta)` is equal to ______.
If 1 + sin2α = 3 sinα cosα, then values of cot α are ______.
Simplify (1 + tan2θ)(1 – sinθ)(1 + sinθ)
(sec2 θ – 1) (cosec2 θ – 1) is equal to ______.
