Write the value of \( \sin\frac{\pi}{12}\sin\frac{5\pi}{12} \)
Solution:
Using identity,
\[
\sin A\sin B
=
\frac12[\cos(A-B)-\cos(A+B)]
\]
\[
\sin\frac{\pi}{12}\sin\frac{5\pi}{12}
\]
\[
=
\frac12
\left[
\cos\left(\frac{\pi}{12}-\frac{5\pi}{12}\right)
–
\cos\left(\frac{\pi}{12}+\frac{5\pi}{12}\right)
\right]
\]
\[
=
\frac12
\left[
\cos\left(-\frac{\pi}{3}\right)
–
\cos\left(\frac{\pi}{2}\right)
\right]
\]
Using,
\[
\cos(-\theta)=\cos\theta
\]
\[
=
\frac12
\left[
\cos\frac{\pi}{3}-0
\right]
\]
\[
=
\frac12\times\frac12
\]
\[
=\frac14
\]
\[
\boxed{\frac14}
\]