JAMIA 2023 — Mathematics PYQ

Step 1: Take the Least Common Multiple (LCM)

Find a common denominator for the two fractions:

Step 2: Simplify the Numerator

Multiply and divide the numerator by $2$:

We know from trigonometric values that $\sin 30^{\circ }=\frac{1}{2}$ and $\cos 30^{\circ }=\frac{\sqrt{3}}{2}$. Substituting these values into the expression:

Using the identity $\sin (A-B)=\sin A\cos B-\cos A\sin B$:

Step 3: Simplify the Denominator

The denominator is $\sin 10^{\circ }\cos 10^{\circ }$. We can use the double-angle formula $\sin 2\theta =2\sin \theta \cos \theta$, which implies $\sin \theta \cos \theta =\frac{1}{2}\sin 2\theta$:

Step 4: Final Calculation

Now, substitute the simplified numerator and denominator back into the main equation:

The term $\sin 20^{\circ }$ cancels out:

Final Answer: