The value of 2( sin)2θ +4 cos(θ + α ) sinα sinθ + cos2 (α + θ ) is

Question

The value of 2( sin)2θ +4 cos(θ + α ) sinα sinθ + cos2 (α + θ ) is (A)  cos θ + cos ⁡ α  (B) independent of θ  (C) independent of α  (D) independent of both θ and α

Tardigrade Admin · Accepted Answer

2 sin 2 θ+2 cos (θ+α)( cos (θ-α)- cos (θ+α))+(2 cos 2(θ+α)-1) =2 sin 2 θ+2( cos 2 θ- sin 2 α)-1 =2( sin )2 θ+2(( cos )2 θ-( sin )2 α)-1 =2-2 sin 2 α-1= cos 2 α- sin 2 α

Q. The value of $2 (sin)^{2} θ + 4 cos (θ + α) sin α sin θ + cos 2 (α + θ)$ is

$cos θ + cos α$

independent of $θ$

independent of $α$

independent of both $θ$ and $α$

$2 sin^{2} θ + 2 cos (θ + α) (cos (θ - α) - cos (θ + α)) + (2 cos^{2} (θ + α) - 1)$
$= 2 sin^{2} θ + 2 (cos^{2} θ - sin^{2} α) - 1$
$= 2 (sin)^{2} θ + 2 ((cos)^{2} θ - (sin)^{2} α) - 1$
$= 2 - 2 sin^{2} α - 1 = cos^{2} α - sin^{2} α$

Q. The value of 2(sin)2θ+4cos(θ+α)sinαsinθ+cos2(α+θ) is

cosθ+cos⁡α

independent of θ

independent of α

independent of both θ and α