Advertisements
Advertisements
प्रश्न
If A + B + C = 180°, prove that sin2A + sin2B − sin2C = 2 sin A sin B cos C
Advertisements
उत्तर
LH.S = `(1 - cos2"A")/2 + (1 - cos2"B")/2 - (1 - cos2"C")/2`
Hint: `[sin^2"A" = (1 - cos2"A")/2]`
= `[1/2 + 1/2 - 1/2] - 1/2 [cos2"A" + cos2"B" - cos 2"C"]`
= `1/2 - 1/2 [2cos("A" + "B") cos("A" - "B") - (2cos^2"C" - 1)]`
= `1/2 - cos("A" + "B") cos("A" - "B") + cos^2"C" - 1/2`
= cos C cos(A – B) + cos2C
= cos C [cos(A – B) – cos(A + B)]
= cos C [2 sin A sin B]
= 2 sin A sin B cos C
= R.H.S
APPEARS IN
संबंधित प्रश्न
Find the value of the trigonometric functions for the following:
cos θ = `- 2/3`, θ lies in the IV quadrant
Find the value of the trigonometric functions for the following:
tan θ = −2, θ lies in the II quadrant
Find the value of the trigonometric functions for the following:
sec θ = `13/5`, θ lies in the IV quadrant
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`
Find the value of sin105°.
Find the value of tan `(7pi)/12`
Prove that sin 105° + cos 105° = cos 45°
Show that tan 75° + cot 75° = 4
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
Show that tan(45° − A) = `(1 - tan "A")/(1 + tan "A")`
Prove that `tan(pi/4 + theta) tan((3pi)/4 + theta)` = – 1
Express the following as a sum or difference
sin 35° cos 28°
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`
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 `tan "A"/2 tan "B"/2 + tan "B"/2 tan "C"/2 + tan "C"/2 tan "A"/2` = 1
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 A + B + C = `pi/2`, prove the following sin 2A + sin 2B + sin 2C = 4 cos A cos B cos C
Choose the correct alternative:
cos 1° + cos 2° + cos 3° + ... + cos 179° =
