Question : If $\sin\left ( 2a+45^{\circ} \right )=\cos\left ( 30^{\circ}-a \right )$ where $0^{\circ}< a< 90^{\circ}$, then the value of a is:
Option 1: $0^{\circ}$
Option 2: $15^{\circ}$
Option 3: $45^{\circ}$
Option 4: $60^{\circ}$
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Correct Answer: $15^{\circ}$
Solution :
$\sin\left ( 2a+45^{\circ} \right )=\cos\left ( 30^{\circ}-a \right )$
⇒ $\sin\left ( 2a+45^{\circ} \right )=\sin\left ( 90^{\circ}–(30^{\circ}-a \right ))$
⇒ $2a+45^{\circ} = 90^{\circ}–(30^{\circ}–a)$
⇒ $2a+45^{\circ} = 90^{\circ}–30^{\circ}+a$
⇒ $a = 60^{\circ}–45^{\circ}$
⇒ $a = 15^{\circ}$
Hence, the correct answer is $15^{\circ}$.
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