If x ∈(π, (3 π/2)), then 4 cos 2((π/4)-(x/2))+√4 sin 4 x+ sin 2 2 x is always equal to

Question

If x ∈(π, (3 π/2)), then 4 cos 2((π/4)-(x/2))+√4 sin 4 x+ sin 2 2 x is always equal to (A) 1 (B) 2 (C) -2 (D) none of these

Tardigrade Admin · Accepted Answer

4 cos 2((π/4)-(x/2))+√4 sin 4 x+ sin 2 2 x =4 cos 2((π/4)-(x/2))+√4 sin 2 x( cos 2 x+ sin 2 x) =2(1+ cos ((π/2)-x))+2| sin x| =2+2 sin x-2 sin x as x ∈(π, (3 π/2))=2

Q. If $x \in (π, \frac{3 π}{2})$ , then $4 cos^{2} (\frac{π}{4} - \frac{x}{2}) + 4 sin^{4} x + sin^{2} 2 x$ is always equal to

1

2

-2

none of these

$4 cos^{2} (\frac{π}{4} - \frac{x}{2}) + 4 sin^{4} x + sin^{2} 2 x$
$= 4 cos^{2} (\frac{π}{4} - \frac{x}{2}) + 4 sin^{2} x (cos^{2} x + sin^{2} x)$
$= 2 (1 + cos (\frac{π}{2} - x)) + 2∣ sin x ∣$
$= 2 + 2 sin x - 2 sin x$ as $x \in (π, \frac{3 π}{2}) = 2$

Q. If x∈(π,23π​), then 4cos2(4π​−2x​)+4sin4x+sin22x​ is always equal to

1

2

-2

none of these

4cos2(4π​−2x​)+4sin4x+sin22x​ =4cos2(4π​−2x​)+4sin2x(cos2x+sin2x)​ =2(1+cos(2π​−x))+2∣sinx∣ =2+2sinx−2sinx as x∈(π,23π​)=2

Q. If $x \in (π, \frac{3 π}{2})$ , then $4 cos^{2} (\frac{π}{4} - \frac{x}{2}) + 4 sin^{4} x + sin^{2} 2 x$ is always equal to

$4 cos^{2} (\frac{π}{4} - \frac{x}{2}) + 4 sin^{4} x + sin^{2} 2 x$
$= 4 cos^{2} (\frac{π}{4} - \frac{x}{2}) + 4 sin^{2} x (cos^{2} x + sin^{2} x)$
$= 2 (1 + cos (\frac{π}{2} - x)) + 2∣ sin x ∣$
$= 2 + 2 sin x - 2 sin x$ as $x \in (π, \frac{3 π}{2}) = 2$