Question 8 - A-Level Maths

By applying de Moivre's theorem to $( \cos \theta + i \sin \theta ) ^ { 4 }$, where $\cos \theta \neq 0$, show that $$( 1 + i \tan \theta ) ^ { 4 } + ( 1 - i \tan \theta ) ^ { 4 } = \frac { 2 \cos 4 \theta } { \cos ^ { 4 } \theta }$$
Hence show that $z = \mathrm { i } \tan \frac { \pi } { 8 }$ satisfies the equation $( 1 + z ) ^ { 4 } + ( 1 - z ) ^ { 4 } = 0$, and express the three other roots of this equation in the form $\mathrm { i } \tan \phi$, where $0 < \phi < \pi$.
Use the results from part (b) to find the values of:
1. $\tan ^ { 2 } \frac { \pi } { 8 } \tan ^ { 2 } \frac { 3 \pi } { 8 }$;
2. $\tan ^ { 2 } \frac { \pi } { 8 } + \tan ^ { 2 } \frac { 3 \pi } { 8 }$.
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