Question : Find the value of $\frac{\sin ^2 39^{\circ}+\sin ^2\left(90^{\circ}–39^{\circ}\right)}{\cos ^2 35^{\circ}+\cos ^2\left(90^{\circ}–35^{\circ}\right)}+3 \tan 25^{\circ} \tan 75^{\circ}$:
Option 1: 2
Option 2: 4
Option 3: 3
Option 4: 1
Correct Answer: 4
Solution :
$\frac{\sin ^2 39^{\circ}+\sin ^2\left(90^{\circ}–39^{\circ}\right)}{\cos ^2 35^{\circ}+\cos ^2\left(90^{\circ}–35^{\circ}\right)}+3 \tan 25^{\circ} \tan 75^{\circ}$
$=\frac{\sin ^2 39^{\circ}+\cos ^239^{\circ}}{\cos ^2 35^{\circ}+\sin ^235^{\circ}}+3 \tan 25^{\circ} \times\tan(90^{\circ}– 25^{\circ})$
$=1+3 \tan 25^{\circ} \times\cot25^{\circ}$
$=1+3$
$=4$
Hence, the correct answer is 4.
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