Question

How do you prove `cos36*cos72=1/4`?

Answer 1

Assuming you do not already know the values of the two cosines, you can find both the difference and the product using the double-angle and sum-product relations for cosine. See the explanation.

Explanation:

First you should prove that

`\cos(36°)-\cos(72°)=1/2`

To do this let `x` be the above difference and take its square:

`x^2=\cos^2(36°)-2\cos(36°)\cos(72°)+\cos^2(72°)`

Apply trigonometric identities you should know. From the double angle formula for cosine:

`\cos^2(36°)+\cos^2(72°)=((1+\cos(72°))/2)+((1+\cos(144°))/2)`

`=1+((\cos(72°)-\cos(36^o))/2)`

`=1-x/2`

Then from the sum-product relation for cosines:

`2\cos(36°)\cos(72°)=\cos(108°)+\cos(36°) =\cos(36°)-\cos(72°)=x`

Put these results together to get:

`x^2=1-(x/2)-x=1-(3/2)x`

Thus

`2x^2-3x-2=0`

This has two roots `x=1/2` and `x=-2`. We know that `x` must be positive so:

`x=\cos(36°)-\cos(72°)=1/2`

Now the product is immediately known because we already calculated:

`2\cos(36°)\cos(72°)=x`

So `\cos(36°)\cos(72°)=1/4`.