Question : If $\tan (x + y)=1$ and $\cos (x-y)={\frac{\sqrt3}{2}}$, then what is the value of $x$ and $y$?
Option 1: $x = 3^\circ, y = 4.5^\circ$
Option 2: $x = 37.5^\circ, y = 7.5^\circ$
Option 3: $x = 7.5^\circ, y = 37.5^\circ$
Option 4: $x = 4.5^\circ, y = 3^\circ$
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Correct Answer: $x = 37.5^\circ, y = 7.5^\circ$
Solution :
$\tan (x + y)=1=\tan 45^\circ$
$⇒x+y=45^\circ$ ____(i)
$\cos (x-y)={\frac{\sqrt3}{2}}=\cos 30^\circ$
$⇒x+y=30^\circ$ ____(ii)
Adding both equations,
$⇒2x=75^\circ$
$⇒x=37.5^\circ$
And,
$⇒2y=15^\circ$
$⇒y=7.5^\circ$
Hence, the correct answer is $x = 37.5^\circ, y = 7.5^\circ$.
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