Prove that : `(2 cos 2^n theta + 1)/(2 cos theta +1) = (2 cos theta -1) (2 cos 2theta -1) (2 cos 2^2 theta -1) … (2 cos 2^(n-1) theta -1)`

Explanation

We have to prove that
`(2cos 2^(n)theta+1)/(2cos theta+1)=(2cos theta-1)(2cos 2theta-1)(2cos2^(2)theta-1)......(2cos 2^(n-1)theta-1)`
or `2cos2^(n)theta+1=[(2cos theta+1)(2cos theta-1)](2cos 2 theta-1)`
`(2cos 2^(2)theta-1).......(2 cos 2^(n-1)theta-1)`
Now `[(2 cos theta+1)(2cos theta-1)](2cos 2^(n-1)theta-1)`
`=(4cos^(2)theta-1)(2cos 2theta-1)(2cos2^(2)theta-1)`......
`(2cos 2^(n-1)theta-1)`
`=(2 cos 2theta+1)(2cos 2theta-1)(2cos 2^(2)theta-1)`........
`(2cos^(n-1)theta-1)` [using `cos 2 theta=2cos^(2)theta-1`]
`=(4cos^(2)2theta-1)(2cos 2^(2)theta-1).....(2cos 2^(n-1)theta-1)`
`=(2cos 2^(2)theta+1)(2cos2^(2)theta-1).........(2cos2^(n-1)theta-1)`
`=(4cos^(2)2^(2)theta-1)(2cos2^(3)theta-1).....(2cos2^(n-1)theta-1)`
`=(2cos2^(n-1)theta+1)(2cos 2^(n-1)theta-1)`
`=4cos^(2)2^(n-1)theta-1`
`=2 cos 2^(n)theta+1`