Advertisements
Advertisements
प्रश्न
If A + B + C = 180°, prove that sin2A + sin2B + sin2C = 2 + 2 cos A cos B cos C
उत्तर
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
संबंधित प्रश्न
Find the value of the trigonometric functions for the following:
sec θ = `13/5`, θ lies in the IV quadrant
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(30° + x) = `(sqrt(3) cos x - sin x)/2`
Expand cos(A + B + C). Hence prove that cos A cos B cos C = sin A sin B cos C + sin B sin C cos A + sin C sin A cos B, if A + B + C = `pi/2`
Prove that sin(30° + θ) + cos(60° + θ) = cos θ
If a cos(x + y) = b cos(x − y), show that (a + b) tan x = (a − b) cot y
Prove that sin 75° – sin 15° = cos 105° + cos 15°
Show that tan 75° + cot 75° = 4
If tan x = `"n"/("n" + 1)` and tan y = `1/(2"n" + 1)`, find tan(x + y)
Prove that `tan(pi/4 + theta) tan((3pi)/4 + theta)` = – 1
Prove that cos 5θ = 16 cos5θ – 20 cos3θ + 5 cos θ
Prove that sin 4α = `4 tan alpha (1 - tan^2alpha)/(1 + tan^2 alpha)^2`
If A + B = 45°, show that (1 + tan A)(1 + tan B) = 2
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 product
cos 65° + cos 15°
If A + B + C = 180°, prove that cos A + cos B − cos C = `- 1 + 4cos "A"/2 cos "B"/2 sin "C"/2`
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`
Choose the correct alternative:
`1/(cos 80^circ) - sqrt(3)/(sin 80^circ)` =
Choose the correct alternative:
If `pi < 2theta < (3pi)/2`, then `sqrt(2 + sqrt(2 + 2cos4theta)` equals to
Choose the correct alternative:
cos 1° + cos 2° + cos 3° + ... + cos 179° =
