Advertisements
Advertisements
Question
If A + B + C = 180°, prove that sin2A + sin2B + sin2C = 2 + 2 cos A cos B cos C
Advertisements
Solution
L.H.S = `(1 - cos2"A")/2 + (1 - cos2"B")/2 + (1 - cos 2"C")/2`
Hint: `[sin^2"A" = (1 - cos2"A")/2]`
= `3/2 - 1/2[cos2"A" + cos2"B" + cos2"C"]`
= `3/2 - 1/2 [2cos("A" + "B") cos("A" - "B") + 2cos^2"C" - 1]`
= `3/2 - cos("A" + "B") cos("A" - "B") - cos^2"C" + 1/2`
= 2 + cos C cos(A – B) – cos2
= 2 + cosC[cos(A – B)(cos(A + B)]
[cos(180° – C) – cos C – cos C]
= 2 + cos C [cos(A – B) + cos(A + B)]
= 2+ cos C[2 cos A cos B]
= 2 + 2 cos A cos B cos C
= R.H.S
APPEARS IN
RELATED QUESTIONS
Find the values of sin(480°)
Find the value of the trigonometric functions for the following:
tan θ = −2, θ lies in the II quadrant
Prove that `(cot(180^circ + theta) sin(90^circ - theta) cos(- theta))/(sin(270^circ + theta) tan(- theta) "cosec"(360^circ + theta))` = cos2θ cotθ
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)
If sin x = `15/17` and cos y = `12/13, 0 < x < pi/2, 0 < y < pi/2`, find the value of cos(x − y)
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)
Prove that cos(π + θ) = − cos θ
Prove that sin(n + 1) θ sin(n – 1) θ + cos(n + 1) θ cos(n – 1)θ = cos 2θ, n ∈ Z
Prove that cos(A + B) cos(A – B) = cos2A – sin2B = cos2B – sin2A
Prove that sin2(A + B) – sin2(A – B) = sin2A sin2B
Prove that `tan(pi/4 + theta) tan((3pi)/4 + theta)` = – 1
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)`
Show that `cot(7 1^circ/2) = sqrt(2) + sqrt(3) + sqrt(4) + sqrt(6)`
Show that sin 12° sin 48° sin 54° = `1/8`
Prove that sin x + sin 2x + sin 3x = sin 2x (1 + 2 cos x)
Prove that `(sin 4x + sin 2x)/(cos 4x + cos 2x)` = tan 3x
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 cos A + cos B − cos C = `- 1 + 4cos "A"/2 cos "B"/2 sin "C"/2`
If A + B + C = `pi/2`, prove the following cos 2A + cos 2B + cos 2C = 1 + 4 sin A sin B sin C
Choose the correct alternative:
cos 1° + cos 2° + cos 3° + ... + cos 179° =
