Advertisements
Advertisements
Question
Prove that `(sin x + sin 3x + sin 5x + sin 7x)/(cos x + cos x + cos 5x cos 7x)` = tan 4x
Solution
Nr: (sin x + sin 7x) + (sin 3x + sin 5x)
= `[2sin (7x + x)/2 cos (7x - x)/2] + [2sin (5x + 3x)/2 cos (5x - 3x)/2]`
= 2 sin 4x cos 3x + 2 sin 4x cos x
= 2 sin 4x (cos 3x + cos x) .....(1)
Dr. (cos x + cos 7x) + (cos 3x + cos 5x)
= `[2cos (7x + x)/2 cos (7x - x)/2] + [2cos (5x + 3x)/2 cos (5x - 3x)/2]`
= 2 cos 4x cos 3x + 2 cos 4x cosx
= 2 cos 4x (cos 3x + cos x) .....(2)
L.H.S = `((1))/((2))`
= `(2sin 4x(cos 3x + cos x))/(2cos 4x(cos 3x + cos x))`
= tan 4x
= R.H.S
APPEARS IN
RELATED QUESTIONS
Find the values of cot(660°)
Find the value of the trigonometric functions for the following:
cos θ = `- 1/2`, θ lies in the III quadrant
Show that `sin^2 pi/18 + sin^2 pi/9 + sin^2 (7pi)/18 + sin^2 (4pi)/9` = 2
Find cos(x − y), given that cos x = `- 4/5` with `pi < x < (3pi)/2` and sin y = `- 24/25` with `pi < y < (3pi)/2`
Prove that sin(π + θ) = − sin θ.
Find a quadratic equation whose roots are sin 15° and cos 15°
Prove that sin(45° + θ) – sin(45° – θ) = `sqrt(2) sin θ`
Prove that sin(30° + θ) + cos(60° + θ) = cos θ
Prove that sin2(A + B) – sin2(A – B) = sin2A sin2B
If cos(α – β) + cos(β – γ) + cos(γ – α) = `- 3/2`, then prove that cos α + cos β + cos γ = sin α + sin β + sin γ = 0
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)`
Prove that `tan (pi/4 + theta) - tan(pi/4 - theta)` = 2 tan 2θ
Express the following as a sum or difference
cos 5θ cos 2θ
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`
Show that `(sin 8x cos x - sin 6x cos 3x)/(cos 2x cos x - sin 3x sin 4x)` = tan 2x
Prove that sin x + sin 2x + sin 3x = sin 2x (1 + 2 cos x)
Prove that 1 + cos 2x + cos 4x + cos 6x = 4 cos x cos 2x cos 3x
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 ∆ABC is a right triangle and if ∠A = `pi/2` then prove that sin2 B + sin2 C = 1
Choose the correct alternative:
If `pi < 2theta < (3pi)/2`, then `sqrt(2 + sqrt(2 + 2cos4theta)` equals to
