Processing Math: Done

From Förberedande kurs i matematik 1

Using the formula for double angles on sin160 gives

sin160=2cos80sin80.

On the right-hand side, we see that the factor cos80 has appeared, and if we use the formula for double angles on the second factor (sin80),

2cos80sin80=2cos802cos40sin40

we obtain a further factor cos40. A final application of the formula for double angles on sin40 gives us all three cosine factors,

2cos802cos40sin40=2cos802cos402cos20sin20·

We have thus succeeded in showing that

sin160=8cos80cos40cos20sin20

which can also be written as

cos80cos40cos20=sin1608sin20.

If we draw the unit circle, we see that 160 makes an angle of 20 with the negative x-axis, and therefore the angles 20 and 160 have the same y-coordinate in the unit circle, i.e.

sin20=sin160.

This shows that

cos80cos40cos20=sin1608sin20=81.