What is expand ((1-cos(2x))/2)^3 ?

The solution to expand ((1-cos(2x))/2)^3 is sin^6(x)

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expand ((1-cos(2x))/2)^3

expand

(\frac{1 - cos ( 2 x )}{2})^{3}

sin^{6} (x)

Solution steps

(\frac{1 - cos ( 2 x )}{2})^{3}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 1 - cos ( 2 x ) ) ^{3}}{2 ^{3}}

(1 - cos (2 x))^{3} = 1 - 3 cos (2 x) + 3 cos^{2} (2 x) - cos^{3} (2 x)

= \frac{1 - 3 cos ( 2 x ) + 3 cos ^{2} ( 2 x ) - cos ^{3} ( 2 x )}{2 ^{3}}

2^{3} = 8

= \frac{1 - 3 cos ( 2 x ) + 3 cos ^{2} ( 2 x ) - cos ^{3} ( 2 x )}{8}

Factor

1 - cos^{3} (2 x) - 3 cos (2 x) + 3 cos^{2} (2 x) : - (cos (2 x) - 1)^{3}

= \frac{- ( - 1 + cos ( 2 x ) ) ^{3}}{8}

Rewrite using trig identities

= sin^{6} (x)

\frac{( a ^{2} b ^{3} c - 4 ) 2}{( ab c ) 3}

\frac{6 x}{( x ^{2} + 2 ) ^{2}}

\frac{( sec ( x ) ) ^{2} β - sin ( β ) tan ( β )}{cos ( x )}

- \frac{1}{s ( s + 2 )}

x - (2 x + 1 = 8 - (3 x + 3))

What is expand ((1-cos(2x))/2)^3 ?
The solution to expand ((1-cos(2x))/2)^3 is sin^6(x)