Advertisements
Advertisements
प्रश्न
Prove the following trigonometric identities.
`(cosec A)/(cosec A - 1) + (cosec A)/(cosec A = 1) = 2 sec^2 A`
उत्तर
We need to prove `(cosec A)/(cosec A - 1) + (cosec A)/(cosec A = 1) = 2 sec^2 A`
Using identity `a^2 - b^2 = (a + b)(a - b)` we get
`(cosec A)/(cosec A - 1) = (cosec A)/(cosec A + 1) = (cosec A(cosec A + 1)+cosec A(cosec A - 1))/(cosec^2 A - 1)`
`= (cosec A (cosec A +1 + cosec A - 1))/(cosec^2 A - 1)`
Further, using the property `1 + cot62 theta = cosec^2 theta` we get
So
`(cosec A (cosec A + 1 + cosec A - 1))/(cosec^2 A- 1) = (cosec A(2 cosec A))/cot^2 A`
`= (2cosec^2 A)/cot^2 A`
`= (2)(1/sin^2 A)((cos^2 A)/(sin^2 A))`
`= 2(1/cos^2 A)`
`= 2 sec^2 A`
Hence proved.
APPEARS IN
संबंधित प्रश्न
`"If "\frac{\cos \alpha }{\cos \beta }=m\text{ and }\frac{\cos \alpha }{\sin \beta }=n " show that " (m^2 + n^2 ) cos^2 β = n^2`
`(1+tan^2A)/(1+cot^2A)` = ______.
Prove the following trigonometric identities.
(1 + tan2θ) (1 − sinθ) (1 + sinθ) = 1
Prove the following trigonometric identities.
`1/(1 + sin A) + 1/(1 - sin A) = 2sec^2 A`
Prove the following trigonometric identities.
`(sec A - tan A)/(sec A + tan A) = (cos^2 A)/(1 + sin A)^2`
`(1 + cot^2 theta ) sin^2 theta =1`
`1/((1+ sintheta ))+1/((1- sin theta ))= 2 sec^2 theta`
Write the value of tan1° tan 2° ........ tan 89° .
Prove that secθ + tanθ =`(costheta)/(1-sintheta)`.
What is the value of \[\frac{\tan^2 \theta - \sec^2 \theta}{\cot^2 \theta - {cosec}^2 \theta}\]
2 (sin6 θ + cos6 θ) − 3 (sin4 θ + cos4 θ) is equal to
Prove the following identity :
cosecθ(1 + cosθ)(cosecθ - cotθ) = 1
Prove the following identity :
`sin^2Acos^2B - cos^2Asin^2B = sin^2A - sin^2B`
Prove the following identity :
`(secA - 1)/(secA + 1) = sin^2A/(1 + cosA)^2`
Prove that:
`sqrt(( secθ - 1)/(secθ + 1)) + sqrt((secθ + 1)/(secθ - 1)) = 2cosecθ`
Prove that `sqrt((1 - sin θ)/(1 + sin θ)) = sec θ - tan θ`.
Prove that cos2θ . (1 + tan2θ) = 1. Complete the activity given below.
Activity:
L.H.S = `square`
= `cos^2theta xx square .....[1 + tan^2theta = square]`
= `(cos theta xx square)^2`
= 12
= 1
= R.H.S
If cosA + cos2A = 1, then sin2A + sin4A = 1.
If 2sin2θ – cos2θ = 2, then find the value of θ.
Prove the following that:
`tan^3θ/(1 + tan^2θ) + cot^3θ/(1 + cot^2θ)` = secθ cosecθ – 2 sinθ cosθ
