Advertisements
Advertisements
Question
Prove the following trigonometric identities
cosec6θ = cot6θ + 3 cot2θ cosec2θ + 1
Solution
We need to prove cosec6θ = cot6θ + 3 cot2θ cosec2θ + 1
Solving the L.H.S, we get
`cosec^6 theta = (cosec^2 theta)^3`
`= (1 + cot^2 theta)^3` .......`(1 + cot^2 theta = cosec^2 theta)`
Further using the identity `(a + b)^3 = a^3 + b^3 + 3a^2b + 3ab^2` we get
`(1 + cot^2 theta)^3 = 1 + cot^6 theta + 3(1)^2 (cot^2 theta) + 3(1) (cot^2 theta)^2`
`= 1 + cot^6 theta + 3 cot^2 theta + 3 cot^4 theta`
`= 1 + cot^6 theta + 3 cot^2 theta (1 + cot^2 theta)`
`= 1 + cot^6 theta + 3 cot^2 theta cosec^2 theta` `(using 1 + cot^2 theta = cosec^2 theta)`
Hence proved.
APPEARS IN
RELATED QUESTIONS
If (secA + tanA)(secB + tanB)(secC + tanC) = (secA – tanA)(secB – tanB)(secC – tanC) prove that each of the side is equal to ±1. We have,
Prove the following trigonometric identities.
`sin^2 A + 1/(1 + tan^2 A) = 1`
Prove the following trigonometric identities.
`sqrt((1 - cos theta)/(1 + cos theta)) = cosec theta - cot theta`
Prove the following trigonometric identities. `(1 - cos A)/(1 + cos A) = (cot A - cosec A)^2`
Prove the following trigonometric identities.
`(tan^2 A)/(1 + tan^2 A) + (cot^2 A)/(1 + cot^2 A) = 1`
Prove the following trigonometric identities.
`(1 + cos theta + sin theta)/(1 + cos theta - sin theta) = (1 + sin theta)/cos theta`
Prove the following trigonometric identities.
`1 + cot^2 theta/(1 + cosec theta) = cosec theta`
Prove the following identities:
`(cotA + cosecA - 1)/(cotA - cosecA + 1) = (1 + cosA)/sinA`
`cosec theta (1+costheta)(cosectheta - cot theta )=1`
`(1+ cos theta - sin^2 theta )/(sin theta (1+ cos theta))= cot theta`
`(sin theta)/((sec theta + tan theta -1)) + cos theta/((cosec theta + cot theta -1))=1`
`(sin theta +cos theta )/(sin theta - cos theta)+(sin theta- cos theta)/(sin theta + cos theta) = 2/((sin^2 theta - cos ^2 theta)) = 2/((2 sin^2 theta -1))`
`(cot^2 theta ( sec theta - 1))/((1+ sin theta))+ (sec^2 theta(sin theta-1))/((1+ sec theta))=0`
If `( sin theta + cos theta ) = sqrt(2) , " prove that " cot theta = ( sqrt(2)+1)`.
Write the value of `(1 + cot^2 theta ) sin^2 theta`.
If `cos theta = 2/3 , "write the value of" ((sec theta -1))/((sec theta +1))`
Prove the following identity :
cosecθ(1 + cosθ)(cosecθ - cotθ) = 1
Prove the following identity :
`(1 + tan^2θ)sinθcosθ = tanθ`
Without using trigonometric identity , show that :
`sin(50^circ + θ) - cos(40^circ - θ) = 0`
Prove the following identity:
(sin2θ – 1)(tan2θ + 1) + 1 = 0
