cos5a.latex

$\cos{5a} = \cos{(3a+2a)}$

$=\cos 3a \cos 2a -\sin 3a \sin 2a$

$=\cos{(2a+a)}\cos 2a -\sin{(2a+a)} \sin 2a$

$=(\cos 2a\cos a-\sin2a\sin a)\cos 2a -(\sin2a \cos a + \cos 2a \sin a) \sin 2a$

$=(\cos 2a\cos a-\sin2a\sin a)\cos 2a -(\sin2a \cos a + \cos 2a \sin a) \sin 2a$

$=((\cos^2a-\sin^2a)\cos a-(2\sin a \cos a)\sin a)(\cos^2a-\sin^2a) -((2\sin a \cos a) \cos a + (\cos^2a-\sin^2a) \sin a) (2\sin a\cos a)$

$=(\cos^3a-\sin^2 a\cos a -2\sin^2a\cos a)(\cos^2a-\sin^2a) -(2\sin a \cos^2a + \cos^2a \sin a - \sin^3 a) (2\sin a \cos a)$

$=(\cos^3a - \sin^2 a \cos a - 2 \sin^2a \cos a)(\cos^2a)-(\cos^3a - \sin^2 a \cos a - 2 \sin^2a \cos a)(\sin^2a) -(2\sin a \cos^2a + \cos^2a \sin a - \sin^3 a) (2\sin a \cos a)$

$=(\cos^5a - \sin^2 a \cos^3 a - 2 \sin^2a \cos^3 a)-(\cos^3a \sin^2a - \sin^4 a \cos a - 2 \sin^4a \cos a) - (4\sin^2 a \cos^3a + 2\cos^3a \sin^2 a - 2\sin^4 a \cos a)$

$=(\cos^5a - \sin^2 a \cos^3 a - 2 \sin^2a \cos^3 a)+(-\cos^3a \sin^2a + \sin^4 a \cos a + 2 \sin^4a \cos a) + (-4 \sin^2 a \cos^3a - 2\cos^3a \sin^2 a + 2\sin^4 a \cos a)$

$\bf =\cos^5a - 10 \sin^2a \cos^3 a + 5 \sin^4a \cos a$ // Another proof

$=\cos^5a - 10 (1 - \cos^2a) \cos^3 a + 5 (1- \cos^2a)^2 \cos a$

$=\cos^5a - 10 \cos^3a + 10 \cos^5a + (5 \cos a- 10 \cos^3a + 5 \cos^5a)$

$\cos{5a}=16\cos^5a - 20 \cos^3a + 5 \cos a$