If α cos 2 3 θ+β cos 4 θ=16 cos 6 θ+9 cos 2 θ is an identity then -

Question

If α cos 2 3 θ+β cos 4 θ=16 cos 6 θ+9 cos 2 θ is an identity then - (A) α=1, β=18 (B) α=1, β=24 (C) α=3, β=24 (D) α=4, β=2

Tardigrade Admin · Accepted Answer

LHS =α(4 cos 3 θ-3 cos θ)2+β cos 4 θ =16 α cos 6 θ-24 α cos 4 θ+9 α cos 2 θ+β cos 4 θ =16 α cos 6 θ+(β-24 α) cos 4 θ+9 α cos 2 θ RHS =16 cos 6 θ+9 cos 2 θ comparing LHS and RHS, we get α=1, β=24

Q. If $α cos^{2} 3 θ + β cos^{4} θ = 16 cos^{6} θ + 9 cos^{2} θ$ is an identity then -

$α = 1, β = 18$

$α = 1, β = 24$

$α = 3, β = 24$

$α = 4, β = 2$

LHS $= α (4 cos^{3} θ - 3 cos θ)^{2} + β cos^{4} θ$
$= 16 α cos^{6} θ - 24 α cos^{4} θ + 9 α cos^{2} θ + β cos^{4} θ$
$= 16 α cos^{6} θ + (β - 24 α) cos^{4} θ + 9 α cos^{2} θ$
RHS $= 16 cos^{6} θ + 9 cos^{2} θ$
comparing LHS and RHS, we get $α = 1, β = 24$

Q. If αcos23θ+βcos4θ=16cos6θ+9cos2θ is an identity then -

α=1,β=18

α=1,β=24

α=3,β=24

α=4,β=2