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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