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Question
The value of \[\sin^2 \frac{5\pi}{12} - \sin^2 \frac{\pi}{12}\]
Options
(a) \[\frac{1}{2}\]
(b) \[\frac{\sqrt{3}}{2}\]
(c) 1
(d) 0
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Solution
(b) \[\frac{\sqrt{3}}{2}\] \[\frac{5\pi}{12} = 75°, \frac{\pi}{12} = 15°\]
\[\sin^2 75° - \sin^2 15° \]
\[ = \sin^2 75 ° - \cos^2 75° \left[ \sin\left( 90° - \theta \right) = \cos\theta \right]\]
\[\text{ Now }, \sin75° = \sin(45° + 30°)\]
\[ = \sin45°\cos30°+ \cos45°\sin30°\]
\[ = \frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2} + \frac{1}{\sqrt{2}} \times \frac{1}{2}\]
\[ = \frac{\sqrt{3} + 1}{2\sqrt{2}}\]
\[\cos75°= \cos(45° + 30°)\]
\[ = \cos45° \cos30°- \sin45°\sin30°\]
\[ = \frac{1}{\sqrt{2}} \times \frac{\sqrt{3}}{2} - \frac{1}{\sqrt{2}} \times \frac{1}{2}\]
\[ = \frac{\sqrt{3} - 1}{2\sqrt{2}}\]
\[\text{ Hence } , \]
\[ \sin^2 75° - \cos^2 75° = \left( \frac{\sqrt{3} + 1}{2\sqrt{2}} \right)^2 - \left( \frac{\sqrt{3} - 1}{2\sqrt{2}} \right)^2 \]
\[ = \frac{3 + 1 + 2\sqrt{3} - 3 - 1 + 2\sqrt{3}}{8}\]
\[ = \frac{4\sqrt{3}}{8}\]
\[ = \frac{\sqrt{3}}{2}\]
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