Question

16X sin? 36 x sin?72​

Answer 1

A key observation is 36+72+72=180. This means

sin72=sin(36+72)=sin36cos72+cos36sin72

Using the identity sin72=2sin36cos36, we get

2sin36cos36=sin36cos72+2sin36cos236

Now notice sin36 appears everywhere, so we can divide by it...

2cos36=cos72+2cos236

Use the identity cos72=2cos236−1, we get

2cos36=4cos236−1

So cos36 is one of the solutions to 2x=4x2−1. Using the quadratic formula, you see that the solutions are

2±

√

20

4

=

1±

√

5

2

But you know that cos36 must be positive, so you know the correct value. Now you can get sin72 from any identity you wish, and multiply.