Advertisements
Advertisements
Question
If A + B + C = `pi/2`, prove the following cos 2A + cos 2B + cos 2C = 1 + 4 sin A sin B sin C
Advertisements
Solution
L.H.S = (cos 2A + cos 2B) + cos 2C
= 2 cos(A + B) cos(A – B) + 1 – 2 sin2C
= 1 + 2 sin C(cos(A – B) – 2 sin2C)
∴ cos(A + B) = cos(90° – C) = sin C
= 1 + 2 sin C [cos(A – B) – sin C]
= 1 + 2 sin C [cos(A – B) – cos(A + B)]
= 1 + 2 sin C [2 sin A sin B]
= 1 + 4 sin A sin B sin C
= R.H.S
APPEARS IN
RELATED QUESTIONS
Find the values of `sin (-(11pi)/3)`
Show that `sin^2 pi/18 + sin^2 pi/9 + sin^2 (7pi)/18 + sin^2 (4pi)/9` = 2
If sin A = `3/5` and cos B = `9/41 0 < "A" < pi/2, 0 < "B" < pi/2`, find the value of sin(A + B)
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 cos(π + θ) = − cos 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`
If a cos(x + y) = b cos(x − y), show that (a + b) tan x = (a − b) cot y
If x cos θ = `y cos (theta + (2pi)/3) = z cos (theta + (4pi)/3)`. find the value of xy + yz + zx
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)`
If θ + Φ = α and tan θ = k tan Φ, then prove that sin(θ – Φ) = `("k" - 1)/("k" + 1)` sin α
Find the value of cos 2A, A lies in the first quadrant, when tan A `16/63`
If θ is an acute angle, then find `cos (pi/4 + theta/2)`, when sin θ = `8/9`
Prove that cos 5θ = 16 cos5θ – 20 cos3θ + 5 cos θ
Prove that `32(sqrt(3)) sin pi/48 cos pi/48 cos pi/24 cos pi/12 cos pi/6` = 3
Express the following as a sum or difference
cos 5θ cos 2θ
Show that sin 12° sin 48° sin 54° = `1/8`
Show that `(sin 8x cos x - sin 6x cos 3x)/(cos 2x cos x - sin 3x sin 4x)` = tan 2x
Show that `((cos theta -cos 3theta)(sin 8theta + sin 2theta))/((sin 5theta - sin theta) (cos 4theta - cos 6theta))` = 1
If ∆ABC is a right triangle and if ∠A = `pi/2` then prove that cos B – cos C = `- 1 + 2sqrt(2) cos "B"/2 sin "C"/2`
