Advertisements
Advertisements
Question
Prove the following identity:
1 + 3cosec2θ cot2θ + cot6θ = cosec6θ
Advertisements
Solution
L.H.S. = 1 + 3cosec2θ cot2θ + cot6θ
= 1 + 3 cosec2θ cot2θ + (cot2θ)3
= 1 + 3 cosec2θ (cosec2θ – 1) + (cosec2θ – 1)3 ...`[(1 + cot^2theta = cosec^2theta),(cot^2theta = cosec^2 theta - 1)]`
= 1 + 3 cosec4θ – 3 cosec2θ + cosec6θ – 3 cosec4θ + 3cosec2θ – 1 ...[(a - b)3 = a3 - 3a2b + 3ab2 - b3]
= `cancel(1) + cancel(3 cosec^4θ) - cancel(3 cosec^2θ) + cosec^6θ - cancel(3 cosec^4 θ) + cancel(3cosec^2 θ) cancel(- 1)`
= cosec6θ
= R.H.S.
APPEARS IN
RELATED QUESTIONS
Evaluate the following:
sin 30° + cos 45° + tan 180°
Evaluate the following :
cosec 45° + cot 45° + tan 0°
If tanθ = `1/2`, evaluate `(2sin theta + 3cos theta)/(4cos theta + 3sin theta)`
Eliminate θ from the following:
x = 3secθ , y = 4tanθ
Eliminate θ from the following :
x = 6cosecθ, y = 8cotθ
Eliminate θ from the following :
x = 4cosθ − 5sinθ, y = 4sinθ + 5cosθ
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 identity:
`tantheta/(sectheta - 1) = (sectheta + 1)/tantheta`
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:
If θ = 60°, then `(1 + tan^2theta)/(2tantheta)` is equal to
Select the correct option from the given alternatives:
If cosecθ + cotθ = `5/2`, then the value of tanθ is
Select the correct option from the given alternatives:
The value of tan1°.tan2°tan3°..... tan89° is equal to
Prove the following:
sin2A cos2B + cos2A sin2B + cos2A cos2B + sin2A sin2B = 1
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:
cos4θ − sin4θ +1= 2cos2θ
Prove the following:
sin8θ − cos8θ = (sin2θ − cos2θ) (1 − 2 sin2θ cos2θ)
Prove the following:
sin6A + cos6A = 1 − 3sin2A + 3 sin4A
Prove the following:
(1 + tanA · tanB)2 + (tanA − tanB)2 = sec2A · sec2B
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 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 ______.
If 5 tan θ = 4. then `(5 sin θ − 3 cos θ)/(5 sin θ + 2 cos θ)` = ______.
