Advertisements
Advertisements
Question
Show that `cos pi/15 cos (2pi)/15 cos (3pi)/15 cos (4pi)/15 cos (5pi)/15 cos (6pi)/15 cos (7pi)/15 = 1/128`
Advertisements
Solution
`(pi/15 = 12^circ)`
L.H.S = cos 12° cos 24° cos 36° cos 48° cos 60° cos 72° cos 84° .....(1)
Consider (we know that)
cos A cos(60° + A) cos(60° – A)
= `cos "A" [cos^2 60^circ - sin^2"A"]`
= `cos "A"[1/4 - (1 - cos^2"A")]`
cos A cos(60° + A) cos(60° – A) = `1/4 cos 3"A"`
= `cos "A"[cos^2"A" - 3/4]`
= `(4cos^3 "A" - 3 cos "A")/4`
cos 12° cos 72° cos 48° = `1/4 cos 3(12^circ)`
= `1/4 cos 36^circ`
= `1/4[(sqrt(5) + 1)/4]`
Similarly cos 24° cos 84° cos 36° = `1/4 cos3 (12^circ)`
= `1/4 cos 72^circ`
= `1/4 cos(90^circ - 18^circ)`
= `1/4 sin 18^circ`
= `1/4[(sqrt(5) - 1)/4]`
(1) ⇒ L.H.S = `1/4[(sqrt(5) + 1)/4] * 1/4[(sqrt(5) - 1)/4] * 1/2`
= `1/4((sqrt(5) + 1)/4 * (sqrt(5) - 1)/4) * 1/2`
= `(5 - 1)/(128 xx 4)`
= `1/128`
APPEARS IN
RELATED QUESTIONS
Find the values of sin (– 1110°)
Find the values of cos(300°)
Find the value of the trigonometric functions for the following:
tan θ = −2, θ lies in the II quadrant
If sin x = `15/17` and cos y = `12/13, 0 < x < pi/2, 0 < y < pi/2` find the value of sin(x + y)
Find sin(x – y), given that sin x = `8/17` with 0 < x < `pi/2`, and cos y = `- 24/25`, x < y < `(3pi)/2`
Prove that cos(30° + x) = `(sqrt(3) cos x - sin x)/2`
If a cos(x + y) = b cos(x − y), show that (a + b) tan x = (a − b) cot y
Prove that sin 105° + cos 105° = cos 45°
Show that cos2 A + cos2 B – 2 cos A cos B cos(A + B) = sin2(A + B)
Find the value of tan(α + β), given that cot α = `1/2`, α ∈ `(pi, (3pi)/2)` and sec β = `- 5/3` β ∈ `(pi/2, pi)`
Find the value of cos 2A, A lies in the first quadrant, when sin A = `4/5`
Prove that sin 4α = `4 tan alpha (1 - tan^2alpha)/(1 + tan^2 alpha)^2`
Prove that (1 + sec 2θ)(1 + sec 4θ) ... (1 + sec 2nθ) = tan 2nθ
Show that `((cos theta -cos 3theta)(sin 8theta + sin 2theta))/((sin 5theta - sin theta) (cos 4theta - cos 6theta))` = 1
Prove that cos(30° – A) cos(30° + A) + cos(45° – A) cos(45° + A) = `cos 2"A" + 1/4`
If A + B + C = 180°, prove that sin2A + sin2B − sin2C = 2 sin A sin B cos C
If A + B + C = 180°, prove that sin A + sin B + sin C = `4 cos "A"/2 cos "B"/2 cos "C"/2`
If A + B + C = 180°, prove that sin(B + C − A) + sin(C + A − B) + sin(A + B − C) = 4 sin A sin B sin C
If x + y + z = xyz, then prove that `(2x)/(1 - x^2) + (2y)/(1 - y^2) + (2z)/(1 - z^2) = (2x)/(1 - x^2) (2y)/(1 - y^2) (2z)/(1 - z^2)`
