Advertisements
Advertisements
प्रश्न
Prove the following:
sin6A + cos6A = 1 − 3sin2A + 3 sin4A
Advertisements
उत्तर
L.H.S. = sin6A + cos6A
= (sin2A)3 + (cos2A)3
= (sin2A+ cos2A)3 - 3 sin2A cos2A (sin2A + cos2A)
... [∵ a3 +b3 = (a+b)3 – 3ab(a+b)]
= (1)3 – 3sin2A cos2A (1)
= 1 – 3sin2A cos2A
= 1 – 3sin2A (1 – sin2A)
= 1 – 3sin2A (– 3sin2A)(– sin2A) ...[Multiply each term in the parentheses by – 3sin2A]
= 1 – 3sin2A (– 3sin2A)(– sin2A)
= 1 – 3sin2A + 3sin4A
= R.H.S.
APPEARS IN
संबंधित प्रश्न
Evaluate the following:
sin 30° + cos 45° + tan 180°
Evaluate the following :
cosec 45° + cot 45° + tan 0°
Evaluate the following :
sin 30° × cos 45° × tan 360°
If tanθ = `1/2`, evaluate `(2sin theta + 3cos theta)/(4cos theta + 3sin theta)`
Eliminate θ from the following :
x = 6cosecθ, y = 8cotθ
Eliminate θ from the following :
x = 4cosθ − 5sinθ, y = 4sinθ + 5cosθ
Eliminate θ from the following :
x = 5 + 6cosecθ, y = 3 + 8cotθ
Eliminate θ from the following:
2x = 3 − 4 tan θ, 3y = 5 + 3 sec θ
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 identities:
`tan^3theta/(1 + tan^2theta) + cot^3theta/(1 + cot^2theta` = secθ cosecθ – 2sinθ cosθ
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:
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:
`1 - sin^2theta/(1 + costheta) + (1 + costheta)/sintheta - sintheta/(1 - costheta)` equals
Select the correct option from the given alternatives:
If cosecθ − cotθ = q, then the value of cot θ is
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:
sin4θ + cos4θ = 1 – 2 sin2θ cos2θ
Prove the following:
2(sin6θ + cos6θ) – 3(sin4θ + cos4θ) + 1 = 0
Prove the following:
`(sin^3theta + cos^3theta)/(sintheta + costheta) + (sin^3theta - cos^3theta)/(sintheta - costheta)` = 2
Prove the following:
(sinθ + cosecθ)2 + (cosθ + secθ)2 = tan2θ + cot2θ + 7
Prove the following:
`("cosec"theta + cottheta + 1)/(cottheta + "cosec" theta - 1) = cottheta/("cosec"theta - 1)`
Prove the following:
`("cosec"theta + cottheta - 1)/( "cosec"theta + cot theta + 1) =(1-sintheta)/costheta`
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 ______.
