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:
tan10° + tan35° + tan10°.tan35° = 1
Prove the following:
(sin 3x + sin x)sin x + (cos 3x – cos x) cos x = 0
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:
`1/(tan3"A" - tan"A") - 1/(cot3"A" - cot"A")` = cot2A
Prove the following:
cos7° cos 14° cos28° cos 56° = `sin68^circ/(16cos83^circ)`
Prove the following:
`(sin^2(-160^circ))/(sin^(2)70^circ) + sin(180^circ - theta)/sintheta` = sec220°
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))`
Prove the following:
`cos(pi/4 + x) + cos(pi/4 - x) = sqrt(2)cosx`
Prove the following:
sin26x − sin24x = sin2x sin10x
Prove the following:
cos22x − cos26x = sin4x sin8x
Prove the following:
`sqrt(3) "cosec"20^circ - sec20^circ` = 4
cos4 θ – sin4 θ is equal to ______.
`sqrt(3) "cosec" 20^circ - sec 20^circ` is equal to ______.
Let α and β be such that π < α – β < 3π. If sin α + sin β = `- 21/65` and cos α + cos β = `-27/65`, then the value of `cos ((α - β))/2` is ______.
If `x + 1/x` = 2 cos θ, then `x^n + 1/x^n` is equal to ______.
`(1 - tan^2(45^circ - A))/(1 + tan^2(45^circ - A))` is equal to ______.
If `sqrt((1 + cos A)/(1 - cos A)) = x/y`, then the value of tan A is ______.
If tan A and tan B are the roots of x2 – ax + b = 0, then the value of sin2(A + B) is ______.
If sin 4A – cos 2A = cos 4A – sin 2A `("where", 0 < A < π/4)`, then the value of tan 4A is ______.
If tan x = sin 45° cos 45° + sin 30°, then x is equal to ______.
If θ is acute and `(cos^2θ)/(cot^2 θ - cos^2 θ)` = 3, then θ is equal to ______.
The value of `(1 + cos π/6)(1 + cos π/3)(1 + cos (2π)/3)(1 + cos (7π)/6)` is equal to ______.
If x sin θ = y cos θ = `(2z tan θ)/(1 - tan^2 θ)`, then 4z2(x2 + y2) is equal to ______.
If `(2 sin α)/({1 + cos α + sin α})` = y, then `({1 - cos α + sin α})/(1 + sin α)` = ______.
`(sin θ + sin 2θ)/(1 + cos θ + cos 2θ)` = ______.
If tan α = `1/7`, tan β = `1/3`, then cos 2α = ______.
The value of cos 6x is equal to ______.
The expression `2 cos π/13. cos (9π)/13 + cos (3π)/13 + cos (5π)/13` is equal to ______.
