Let and be real numbers such that 4 . If sin = 1 3 and cos = 2 3 , then the greatest integer less than or equal to ( sin cos + cos sin + cos sin + sin cos ) 2 is _______. [2022]

Question

Let $α$ and $β$ be real numbers such that $- \frac{π}{4} < β < 0 < α < \frac{π}{4}$ . If $\sin (α + β) = \frac{1}{3}$ and $\cos (α - β) = \frac{2}{3}$ ,

then the greatest integer less than or equal to ${(\frac{\sin α}{\cos β} + \frac{\cos β}{\sin α} + \frac{\cos α}{\sin β} + \frac{\sin β}{\cos α})}^{2}$ is _______. [2022]

Smart Achievers · Accepted Answer

(1)

Rearrange the given expression

${(\frac{\sin α}{\cos β} + \frac{\cos α}{\sin β} + \frac{\cos β}{\sin α} + \frac{\sin β}{\cos α})}^{2}$

$= {(\frac{\cos (α - β)}{\sin β \cos β} + \frac{\cos (α - β)}{\sin α . \cos α})}^{2}$

$= {(\frac{4}{3} {\frac{1}{\sin 2 β} + \frac{1}{\sin 2 α}})}^{2}$ $[∵ \cos (α - β) = \frac{2}{3}]$

$= \frac{16}{9} {[\frac{\sin 2 α + \sin 2 β}{\sin 2 α \cdot \sin 2 β}]}^{2}$

$= \frac{64}{9} {(\frac{2 \sin (α + β) . \cos (α - β)}{2 \sin 2 α \cdot \sin 2 β})}^{2}$

$= \frac{64}{9} {(\frac{2 \cdot \frac{1}{3} \cdot \frac{2}{3}}{\cos (2 α - 2 β) - \cos (2 α + 2 β)})}^{2}$

$= \frac{64}{9} {(\frac{\frac{4}{9}}{2 \cos^{2} (α - β) - 1 - 1 + 2 \sin^{2} (α + β)})}^{2}$

$= \frac{64}{9} {(\frac{\frac{4}{9}}{\frac{8}{9} - 2 + \frac{2}{9}})}^{2}$ $= \frac{64}{9} {(- \frac{1}{2})}^{2}$ $= [\frac{16}{9}] = [1.7] = 1$

Solution

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Solution

Q. Let α and β be real numbers such that -π4<β<0<α<π4. If sin(α+β)=13 and cos(α-β)=23, then the greatest integer less than or equal to (sinαcosβ+cosβsinα+cosαsinβ+sinβcosα)2 is _______. [2022]

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