Find the Value of sin²(5π/12) − sin²(π/12)

Solution

We use the identity:

\[ \sin^2 A-\sin^2 B = (\sin A-\sin B)(\sin A+\sin B) \]

Also,

\[ \sin^2 A-\sin^2 B = \sin(A+B)\sin(A-B) \]

Let

\[ A=\frac{5\pi}{12}, \qquad B=\frac{\pi}{12} \]

Then,

\[ A+B = \frac{5\pi}{12}+\frac{\pi}{12} = \frac{6\pi}{12} = \frac{\pi}{2} \]

and

\[ A-B = \frac{5\pi}{12}-\frac{\pi}{12} = \frac{4\pi}{12} = \frac{\pi}{3} \]

Therefore,

\[ \sin^2\left(\frac{5\pi}{12}\right) – \sin^2\left(\frac{\pi}{12}\right) = \sin\frac{\pi}{2}\sin\frac{\pi}{3} \]

\[ = 1\times\frac{\sqrt{3}}{2} \]

\[ = \frac{\sqrt{3}}{2} \]

Final Answer

\[ \boxed{ \sin^2\left(\frac{5\pi}{12}\right) – \sin^2\left(\frac{\pi}{12}\right) = \frac{\sqrt{3}}{2} } \]

Correct Option: (b)