Question

sin2A +sin2B + sin2C =4sinA SinBSinC​

Answer 1

Appropriate Question :-

$\sf \: If \: A + B + C = \pi$

Prove that

$\sf \: sin2A + sin2B + sin2C = 4sinAsinBsinC$

Identities Used :-

$\boxed{ \red{ \bf \: sinx + siny = 2sin\bigg(\dfrac{x + y}{2}\bigg) cos\bigg(\dfrac{x - y}{2} \bigg)}}$

$\boxed{ \red{ \bf \:sin(\pi - x) = sinx}}$

$\boxed{ \red{ \bf \:cos(x - y) - cos(x + y) = 2sinxsiny}}$

$\boxed{ \red{ \bf \:sin2x = 2sinxcosx}}$

$\large\underline{\bf{Solution-}}$

$\rm :\longmapsto\:\sf \: sin2A + sin2B + sin2C$

$\sf \: = \: \: \:(sin2A + sin2B) + sin2C$

$\sf \: = \: \: \:2sin\bigg(\dfrac{2A + AB}{2} \bigg)cos\bigg(\dfrac{2A - 2B}{2} \bigg) + sin2C$

$\sf \: = \: \: \:2sin(A + B)cos(A - B) + sin2C$

$\sf \: = \: \: \:2sin(\pi - C)cos(A - B) + sin2C$

$\sf \: = \: \: \:2sinCcos(A - B) + 2sinCcosC$

$\sf \: = \: \: \:2sinC\bigg(cos(A - B) + cosC \bigg)$

$\sf \: = \: \: \:2sinC\bigg(cos(A - B) + cos(\pi - (A + B))\bigg)$

$\sf \: = \: \: \:2sinC\bigg(cos(A - B) - cos(A + B)\bigg)$

$\sf \: = \: \: \:2sinC(2sinAsinB)$

$\sf \: = \: \: \:4sinCsinAsinB$

$\sf \: = \: \: \:4sinAsinBsinC$

Hence,

$\purple{\boxed{{ \bf \:sin2A + sin2B + sin2C = 4sinAsinBsinC}}}$

Additional Information :-

Trigonometry Formulas

sin(−θ) = −sin θ

cos(−θ) = cos θ

tan(−θ) = −tan θ

cosec(−θ) = −cosecθ

sec(−θ) = sec θ

cot(−θ) = −cot θ

Product to Sum Formulas

sin x sin y = 1/2 [cos(x–y) − cos(x+y)]

cos x cos y = 1/2[cos(x–y) + cos(x+y)]

sin x cos y = 1/2[sin(x+y) + sin(x−y)]

cos x sin y = 1/2[sin(x+y) – sin(x−y)]

Sum to Product Formulas

sin x + sin y = 2 sin [(x+y)/2] cos [(x-y)/2]

sin x – sin y = 2 cos [(x+y)/2] sin [(x-y)/2]

cos x + cos y = 2 cos [(x+y)/2] cos [(x-y)/2]

cos x – cos y = -2 sin [(x+y)/2] sin [(x-y)/2]

Sum or Difference of angles

cos (A + B) = cos A cos B – sin A sin B

cos (A – B) = cos A cos B + sin A sin B

sin (A+B) = sin A cos B + cos A sin B

sin (A -B) = sin A cos B – cos A sin B

tan(A+B) = [(tan A + tan B)/(1 – tan A tan B)]

tan(A-B) = [(tan A – tan B)/(1 + tan A tan B)]

cot(A+B) = [(cot A cot B − 1)/(cot B + cot A)]

cot(A-B) = [(cot A cot B + 1)/(cot B – cot A)]

cos(A+B) cos(A–B)=cos^2A–sin^2B=cos^2B–sin^2A

sin(A+B) sin(A–B) = sin^2A–sin^2B=cos^2B–cos^2A

Multiple and Submultiple angles

sin2A = 2sinA cosA = [2tan A /(1+tan²A)]

cos2A = cos²A–sin²A = 1–2sin²A = 2cos²A–1= [(1-tan²A)/(1+tan²A)]

tan 2A = (2 tan A)/(1-tan²A)