If α ∈( cos -1 (√5-1/2), (π/2)) then which one of the following inequalities hold good?

Question

If α ∈( cos -1 (√5-1/2), (π/2)) then which one of the following inequalities hold good? (A)  cos α< sin α< cot α< tan α (B)  cos α< cot α< sin α< tan α (C)  sin α< tan α< cos α< cot α (D)  sin α< cot α< tan α< cos α

Tardigrade Admin · Accepted Answer

As α ∈( cos -1 (√5-1/2), (π/2)) so cos α ∈(0, (√5-1/2)) [Online test-4, P-1] Now ( cos α- cot α)= cos α(( sin α-1/ sin α))<0 ⇒ cos α< cot α Also ( sin α- tan α)= sin α(( cos α-1/ cos α))<0 ⇒ sin α< tan α And ( cot α- sin α)=(1/ sin α)( cos α- sin 2 α)=(1/ sin α)( cos α-1+ cos 2 α)=(1/ sin α)(( cos α+(1/2))2-(5/4)) =(1/ sin α)( cos α-(√5-1/2))( cos α+(√5-1/2))<0 ∀ cos α ∈(0, (√5-1/2)) ∴( cot α- sin α)<0 ⇒ cot α< sin α ∴ From inequation (1), (2) and (3), we get cos α< cot α< sin α< tan α.

Q. If $α \in (cos^{- 1} \frac{5 - 1}{2}, \frac{π}{2})$ then which one of the following inequalities hold good?

$cos α < sin α < cot α < tan α$

$cos α < cot α < sin α < tan α$

$sin α < tan α < cos α < cot α$

$sin α < cot α < tan α < cos α$

Q. If α∈(cos−125​−1​,2π​) then which one of the following inequalities hold good?

cosα<sinα<cotα<tanα

cosα<cotα<sinα<tanα

sinα<tanα<cosα<cotα

sinα<cotα<tanα<cosα

Q. If $α \in (cos^{- 1} \frac{5 - 1}{2}, \frac{π}{2})$ then which one of the following inequalities hold good?

$cos α < sin α < cot α < tan α$

$cos α < cot α < sin α < tan α$

$sin α < tan α < cos α < cot α$

$sin α < cot α < tan α < cos α$