If α+β+γ=π, then the value of sin 2 α+ sin 2 β- sin 2 γ is equal to

Question

If α+β+&gamma;=π, then the value of sin 2 α+ sin 2 β- sin 2 &gamma; is equal to (A)  2 sin α  (B)  2 sin α cosβ sin&gamma;  (C)  2 sin α sin β cos &gamma;  (D)  2 sin α sin β sin &gamma;

Tardigrade Admin · Accepted Answer

∴ sin 2 α+ sin (β-&gamma;) sin (β+&gamma;) = sin 2 α+ sin (π-α) sin (β-&gamma;) = sin α[ sin α+ sin (β-&gamma;)] = sin α[ sin (π-(β+&gamma;))+ sin (β-&gamma;)] = sin α[ sin (β-&gamma;)+ sin (β+&gamma;)] =2 sin α sin β cos &gamma;

Q. If $α + β + γ = π$ , then the value of $sin^{2} α + sin^{2} β - sin^{2} γ$ is equal to

$2 sin α$

$2 sin α cos β sin γ$

$2 sin α sin β cos γ$

$2 sin α sin β sin γ$

$∴ sin^{2} α + sin (β - γ) sin (β + γ)$
$= sin^{2} α + sin (π - α) sin (β - γ)$
$= sin α [sin α + sin (β - γ)]$
$= sin α [sin (π - (β + γ)) + sin (β - γ)]$
$= sin α [sin (β - γ) + sin (β + γ)]$
$= 2 sin α sin β cos γ$

Q. If α+β+γ=π, then the value of sin2α+sin2β−sin2γ is equal to

2sinα

2sinαcosβsinγ

2sinαsinβcosγ

2sinαsinβsinγ