If $[cos | α |] = {sin x}, α \in R,$ where ${.}$ is the fractional part function and $[.]$ is the greatest integer function, then the value of $5 - log tan ∣ ∣ ∣ \frac{π}{4} - x ∣ ∣ ∣$ is

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Solution

We know that $sin x \in [- 1, 1]$ and ${x} \in [0, 1)$
and $[cos | α |] \in {- 1, 0, 1}$
$∴ [cos | α |] = {sin x}$ is possible only if
${sin x} = 0$ and $α \neq n π, n \in Z$
$∴ x = n π$ and $α \neq n π, n \in Z$

$∴ log tan ∣ ∣ ∣ \frac{π}{4} - x ∣ ∣ ∣ = log tan ∣ ∣ ∣ \frac{π}{4} - n π ∣ ∣ ∣ = log tan \frac{π}{4} = 0$
$\Rightarrow 5 - log tan ∣ ∣ ∣ \frac{π}{4} - x ∣ ∣ ∣ = 5$

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If [cos|α|]={sinx},α∈R, where {.} is the fractional part function and [.] is the greatest integer function, then the value of 5−logtan∣∣∣π4−x∣∣∣ is

We know that sinx∈[−1,1] and {x}∈[0,1) and [cos|α|]∈{−1,0,1} ∴[cos|α|]={sinx} is possible only if {sinx}=0 and α≠nπ, n∈Z ∴x=nπ and α≠nπ, n∈Z ∴logtan∣∣∣π4−x∣∣∣=logtan∣∣∣π4−nπ∣∣∣=logtanπ4=0 ⇒5−logtan∣∣∣π4−x∣∣∣=5

If $[cos | α |] = {sin x}, α \in R,$ where ${.}$ is the fractional part function and $[.]$ is the greatest integer function, then the value of $5 - log tan ∣ ∣ ∣ \frac{π}{4} - x ∣ ∣ ∣$ is