Question

The value of `sin  π/16 sin  (3π)/16 sin  (5π)/16 sin  (7π)/16` is ______.

Options

• `1/16`

• `sqrt(2)/16`

• `1/8`

• `sqrt(2)/8`

Answer 1

The value of `sin  π/16 sin  (3π)/16 sin  (5π)/16 sin  (7π)/16` is `underlinebb(sqrt(2)/16)`.

Explanation:

We have, `sin  π/16 . sin  (3π)/16 . sin  (5π)/16 . sin  (7π)/16`

= `1/2[2 sin  π/16 sin  (3π)/16 . 2 sin  (5π)/16 sin  (7π)/16]`

= `1/4[(cos  π/8 - cos  π/4)(cos  π/8 - cos  (3π)/4)]`

= `1/4[(cos  π/8 - 1/sqrt(2))(cos  π/8 + 1/sqrt(2))]`

= `1/4[(cos^2  π/8 - 1/sqrt(2))]`

= `1/8[2 cos^2  π/8 - 1]`

= `1/8[cos  π/4]`

= `1/8 xx 1/sqrt(2)`

= `sqrt(2)/16`