Question

square root of [(1 - cos8x)/(1+ cos4x)] = ?

Answer 1

\[1-cos( 8x) = 1-cos 2(4x) = 2sin^2(4x) = 2(1-cos^2(4x) = 2(1+cos (4x))(1-cos (4x))\]
now, put it into the square root
\[\sqrt{\frac{2(1+cos(4x))(1-cos(4x))}{1+cos(4x)}}=\sqrt{2(1+cos(4x))}\]
\[1+cos(4x)= 2cos^2(2x)\]
therefore, it turns to \(\sqrt{2*2cos^2(2x)}\)=2cos(2x)