Advertisements
Advertisements
प्रश्न
Prove the following:
cos2x + cos2(x + 120°) + cos2(x – 120°) = `3/2`
उत्तर
L.H.S. = cos2x + cos2(x + 120°) + cos2(x – 120°)
= `(1 + cos 2x)/2 + (1 + cos2(x + 120^circ))/2 + (1 + cos2(x - 120^circ))/2` ......`[∵ cos^2θ = (1+cos2θ)/2]`
`=3/2 + 1/2[cos2x + cos(2x + 240^circ)+cos(2x-240°)]`
= `3/2 + 1/2(cos2x + cos2x cos 240^circ-sin2x sin240°+cos2x cos240°+sin2x sin240°)`
= `3/2+1/2(cos2x+2cos2x cos240°)`
= `3/2 + 1/2[cos 2x +2 cos2x cos(180° + 60°)]`
= `3/2 + 1/2[cos2x + 2cos2x(- cos 60^circ)]`
= `3/2 + 1/2[cos2x - 2cos2x (1/2)]`
= `3/2+1/2(cos2x-cos2x)`
= `3/2+1/2(0)`
= `3/2`
= R.H.S.
APPEARS IN
संबंधित प्रश्न
Prove the following:
`(cos27^circ + sin27^circ)/(cos27^circ - sin27^circ)` = tan72°
Prove the following:
tan10° + tan35° + tan10°.tan35° = 1
Prove the following:
`(cot"A"cot4"A" + 1)/(cot"A" cot4"A" - 1) = (cos3"A")/(cos5"A")`
Prove the following:
(cos x + cos y)2 + (sin x – sin y)2 = `4cos^2 ((x + y))/2`
Prove the following:
(cos x – cos y)2 + (sin x – sin y)2 = `4sin^2 ((x - y))/2`
Prove the following:
16 sin θ cos θ cos 2θ cos 4θ cos 8θ = sin 16θ
Prove the following:
`(tan(theta/2) + cot(theta/2))/(cot(theta/2) - tan(theta/2))` = secθ
Prove the following:
cos7° cos 14° cos28° cos 56° = `sin68^circ/(16cos83^circ)`
Prove the following:
`(2cos 4x + 1)/(2cosx + 1)` = (2 cos x – 1) (2 cos 2x – 1)
Prove the following:
2cosec 2x + cosec x = `secx cot(x/2)`
Prove the following:
`4 cos x. cos(x + pi/3) . cos (x - pi/3)` = cos 3x
Prove the following:
`sinx tan(x/2) + 2cosx = 2/(1 + tan^2(x/2))`
Select the correct option from the given alternatives :
The value of `sin pi/14sin (3pi)/14sin (5pi)/14sin (7pi)/14sin (9pi)/14sin (11pi)/14sin (13pi)/14` is ....
Select the correct option from the given alternatives :
If α + β + γ = π then the value of sin2α + sin2β – sin2γ is equal to …......
Prove the following:
`cos(pi/4 + x) + cos(pi/4 - x) = sqrt(2)cosx`
Prove the following:
`(sin5x - 2sin3x + sinx)/(cos5x - cosx)` = tanx
Prove the following:
cot4x (sin5x + sin3x) = cotx (sin5x − sin3x)
cos4 θ – sin4 θ is equal to ______.
If cos 2α = `(3 cos 2β - 1)/(3 - cos 2β)`, then tan α is equal to ______.
If sin 4A – cos 2A = cos 4A – sin 2A `("where", 0 < A < π/4)`, then the value of tan 4A is ______.
The value of `(cos^3θ - cos 3θ)/cosθ + (sin^3θ + sin 3θ)/sinθ` is ______.
cot x . cot 2x – cot 2x . cot 3x – cot 3x . cot x is equal to ______.
If θ is acute and `(cos^2θ)/(cot^2 θ - cos^2 θ)` = 3, then θ is equal to ______.
If sin θ = `12/13, (0 < θ < π/2)` and cos `phi = - 3/5, (π < phi < (3π)/2)`. Then, sin(θ + `phi`) will be ______.
If `(2 sin α)/({1 + cos α + sin α})` = y, then `({1 - cos α + sin α})/(1 + sin α)` = ______.
If tan β = cos θ tan α, then `tan^2 θ/2` = ______.
2 sin A cos3 A – 2 sin3 A cos A = ______.
The value of `sin π/10` is ______.
