Question

Express cos^7θ in terms of multiples of angles.

Answer 1

According to Euler's formula:

$z^{n}+z^{-n} = 2cosnx$

$128 \cdot cos^{7}x = (2 \cdot cosx)^{7} = (z + z^{-1})^{7}$

Thus, applying binomial formula:

$128 \cdot cos^{7}x = z^{7} + 7\cdot z^{5} + 21\cdot z^{3} + 35\cdot {z} + 35\cdot z^{-1} + 21\cdot z^{-3} + 7\cdot z^{-5}+z^{-7} = 2\cdot cos7x + 14\cdot cos5x + 42\cdot cos3x+70\cdot cosx$

Thus, $cos^{7}x = \frac{2\cdot cos7x + 14\cdot cos5x + 42\cdot cos3x+70\cdot cosx}{128} = \frac{cos7x +7\cdot cos5x + 21\cdot cos3x+35\cdot cosx}{64}$