Advertisements
Advertisements
Question
Prove the following:
`4 cos x. cos(x + pi/3) . cos (x - pi/3)` = cos 3x
Solution
`4 cos x. cos(x + pi/3). cos (x - pi/3)`
Cosine Addition Formula: cos (a+b) = cosa cosb − sina sinb
Cosine Subtraction Formula: cos (a−b) = cosa cosb + sina sinb
Here, a = x and b = `pi/3`
L.H.S. = `4 cos x. cos(x + pi/3). cos (x - pi/3)`
= `4 cosx. (cos x cos pi/3 - sin x sin pi/3) (cosx cos pi/3 + sinx sin pi/3)`
= `4 cosx (cosx 1/2 - sinx sqrt3/2) (cosx 1/2 + sinx sqrt3/2)` ...`[∵ sin pi/3 = sqrt3/2, cos pi/3 = 1/2]`
= `4 cos (1/2 cosx - sqrt3/2 sinx) (1/2 cosx + sqrt3/2 sinx)`
= `4 cosx[(1/2 cosx)^2 - (sqrt3/2 sin x)^2]`
= `4 cosx (1/4cos^2x - 3/4 sin^2)`
= `4/4(cos^3x - 3 sin^2.cosx)`
= `cos^3x - 3 cosx.sin^2x`
= cos3x - 3 cosx (1 - cos2x)
= cos3x - 3 cosx + 3 cos3x
= 4 cos3x - 3 cosx
= cos 3x
= R.H.S.
APPEARS IN
RELATED QUESTIONS
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:
`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:
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:
`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:
`(sin5x - 2sin3x + sinx)/(cos5x - cosx)` = tanx
Prove the following:
sin26x − sin24x = sin2x sin10x
Prove the following:
cos22x − cos26x = sin4x sin8x
Prove the following:
cot4x (sin5x + sin3x) = cotx (sin5x − sin3x)
`sqrt(3) "cosec" 20^circ - sec 20^circ` is equal to ______.
If cos 2α = `(3 cos 2β - 1)/(3 - cos 2β)`, then tan α 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 ______.
The value of `(cos^3θ - cos 3θ)/cosθ + (sin^3θ + sin 3θ)/sinθ` is ______.
If tan x = sin 45° cos 45° + sin 30°, then x 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 ______.
`(sin 3θ - cos 3θ)/(sin θ + cos θ) + 1` = ______.
(sec 2A + 1) sec2 A = ______.
2 sin A cos3 A – 2 sin3 A cos A = ______.
The value of `sin π/10` is ______.
