Advertisements
Advertisements
प्रश्न
Prove the following:
cos7° cos14° cos28° cos56° = `sin68^circ/(16cos83^circ)`
Advertisements
उत्तर
L.H.S. = cos7° cos 14° cos28° cos 56°
= `1/(2sin7^circ)(2sin 7^circcos 7^circ)cos 14^circ cos 28^circ cos 56^circ`
= `1/(2sin7^circ)(sin 14^circ cos 14^circ cos 28^circ cos 56^circ)` ...[∵ 2 sin θ cos θ = sin 2θ]
= `1/(2(2sin 7^circ))(2 sin 14^circ cos 14^circ)cos 28^circ cos 56^circ`
= `1/(4sin7^circ)(sin 28^circ cos 28^circ cos 56^circ)`
= `1/(2(4sin7^circ))(2sin 28^circ cos 28^circ) cos 56^circ`
= `1/(8sin 7^circ)(sin 56^circ cos 56^circ)`
= `1/(2(8sin 7^circ))(2sin 56^circ cos 56^circ)`
= `1/(16 sin 7^circ)(sin112^circ)`
= `(sin(180^circ - 68^circ))/(16sin(90^circ - 83^circ))`
= `(sin 68^circ)/(16cos 83^circ)`
= 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:
`cosx/(1 + sinx) = (cot(x/2) - 1)/(cot(x/2) + 1)`
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:
`(sin^2(-160^circ))/(sin^(2)70^circ) + sin(180^circ - theta)/sintheta` = sec220°
Prove the following:
cos2x + cos2(x + 120°) + cos2(x – 120°) = `3/2`
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:
`(sin5x - 2sin3x + sinx)/(cos5x - cosx)` = tanx
Prove the following:
cos22x − cos26x = sin4x sin8x
Prove the following:
cot4x (sin5x + sin3x) = cotx (sin5x − sin3x)
cos4 θ – sin4 θ 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 ______.
`(1 - tan^2(45^circ - A))/(1 + tan^2(45^circ - A))` is equal to ______.
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 ______.
For any angle θ, the expression `(2 cos 8θ + 1)/(2 cos θ + 1)` is equal to ______.
The value of `(cos^3θ - cos 3θ)/cosθ + (sin^3θ + sin 3θ)/sinθ` is ______.
`(sin(90^circ - θ) sin θ)/tanθ + sin^2 θ` is equal to ______.
If sin θ = `1/2` and θ is acute, then (3 cos θ – 4 cos3 θ) 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 `tan x + tan(π/3 - x) tan ((2π)/3 + x)` = 3, then ______.
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 = ______.
If tan β = cos θ tan α, then `tan^2 θ/2` = ______.
(sec 2A + 1) sec2 A = ______.
2 sin A cos3 A – 2 sin3 A cos A = ______.
The value of `sin π/10` is ______.
The value of sin 3A sin3 A + cos 3A cos3 A is ______.
