If sin x+ sin 2 x=1, then cos 6 x+ cos 12 x+3 cos 10 x+3 cos 8 x is equal to :

Question

If sin x+ sin 2 x=1, then cos 6 x+ cos 12 x+3 cos 10 x+3 cos 8 x is equal to : (A) 1 (B)  cos 3 x sin 3 x (C) 0 (D) ∞

Tardigrade Admin · Accepted Answer

Given sin x+ sin 2 x=1 ⇒ sin x=1- sin 2 x= cos 2 x ⇒ sin x= cos 2 x ∴ cos 6 x+ cos 12 x+3 cos 10 x+3 cos 8 x = sin 3 x+ sin 6 x+3 sin 5 x+3 sin 4 x =( sin x+ sin 2 x)3=13=1

Q. If $sin x + sin^{2} x = 1$ , then $cos^{6} x + cos^{12} x + 3 cos^{10} x + 3 cos^{8} x$ is equal to :

1

$cos^{3} x sin^{3} x$

0

$\infty$

Given $sin x + sin^{2} x = 1$
$\Rightarrow sin x = 1 - sin^{2} x = cos^{2} x$
$\Rightarrow sin x = cos^{2} x$
$∴ cos^{6} x + cos^{12} x + 3 cos^{10} x + 3 cos^{8} x$
$= sin^{3} x + sin^{6} x + 3 sin^{5} x + 3 sin^{4} x$
$= (sin x + sin^{2} x)^{3} = 1^{3} = 1$

Q. If sinx+sin2x=1, then cos6x+cos12x+3cos10x+3cos8x is equal to :

1

cos3xsin3x

0

∞

Given sinx+sin2x=1 ⇒sinx=1−sin2x=cos2x ⇒sinx=cos2x ∴cos6x+cos12x+3cos10x+3cos8x =sin3x+sin6x+3sin5x+3sin4x =(sinx+sin2x)3=13=1

Q. If $sin x + sin^{2} x = 1$ , then $cos^{6} x + cos^{12} x + 3 cos^{10} x + 3 cos^{8} x$ is equal to :

$cos^{3} x sin^{3} x$

$\infty$

Given $sin x + sin^{2} x = 1$
$\Rightarrow sin x = 1 - sin^{2} x = cos^{2} x$
$\Rightarrow sin x = cos^{2} x$
$∴ cos^{6} x + cos^{12} x + 3 cos^{10} x + 3 cos^{8} x$
$= sin^{3} x + sin^{6} x + 3 sin^{5} x + 3 sin^{4} x$
$= (sin x + sin^{2} x)^{3} = 1^{3} = 1$