Question : What is the value of $\frac{4}{3}\left(\sin ^2 35^{\circ}+\sin ^2 55^{\circ}\right) ?$
Option 1: $\frac{2}{3}$
Option 2: $\frac{4}{3}$
Option 3: $\frac{8}{9}$
Option 4: $\frac{3}{4}$
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Correct Answer: $\frac{4}{3}$
Solution :
$\frac{4}{3}(\sin^2 35^{\circ}+\sin^2 55^{\circ})$
$=\frac{4}{3}(\sin^2 35^{\circ}+\sin^2(90^{\circ}-35^{\circ}))$
$=\frac{4}{3}(\sin^2 35^{\circ}+\cos^235^\circ)$
$=\frac{4}{3}\times1$
$=\frac{4}{3}$
Hence, the correct answer is $\frac{4}{3}$.
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