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cos(pi/5)cos((2pi)/5)cos((4pi)/5)cos((8pi)/5)

Solution

cos (\frac{π}{5}) cos (\frac{2 π}{5}) cos (\frac{4 π}{5}) cos (\frac{8 π}{5})

\frac{cos ( \frac{π}{5} ) cos ( \frac{2 π}{5} ) cos ( \frac{4 π}{5} ) + cos ( \frac{π}{5} ) cos ^{2} ( \frac{2 π}{5} )}{2}

Decimal

- 0.0625

Solution steps

cos (\frac{π}{5}) cos (\frac{2 π}{5}) cos (\frac{4 π}{5}) cos (\frac{8 π}{5})

Rewrite using trig identities:

\frac{cos ( \frac{π}{5} ) cos ( \frac{2 π}{5} ) ( cos ( \frac{4 π}{5} ) + cos ( \frac{2 π}{5} ) )}{2}

cos (\frac{π}{5}) cos (\frac{2 π}{5}) cos (\frac{4 π}{5}) cos (\frac{8 π}{5})

Use the Product to Sum identity:

cos (s) cos (t) = \frac{1}{2} (cos (s - t) + cos (s + t))

= cos (\frac{π}{5}) cos (\frac{2 π}{5}) \frac{1}{2} (cos (\frac{4 π}{5} - \frac{8 π}{5}) + cos (\frac{4 π}{5} + \frac{8 π}{5}))

Simplify:

\frac{4 π}{5} - \frac{8 π}{5} = - \frac{4 π}{5}

\frac{4 π}{5} - \frac{8 π}{5}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{4 π - 8 π}{5}

Add similar elements:

4 π - 8 π = - 4 π

= \frac{- 4 π}{5}

Apply the fraction rule:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{4 π}{5}

Simplify:

\frac{4 π}{5} + \frac{8 π}{5} = \frac{12 π}{5}

\frac{4 π}{5} + \frac{8 π}{5}

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{4 π + 8 π}{5}

Add similar elements:

4 π + 8 π = 12 π

= \frac{12 π}{5}

cos (\frac{π}{5}) cos (\frac{2 π}{5}) \frac{1}{2} (cos (- \frac{4 π}{5}) + cos (\frac{12 π}{5})) = \frac{cos ( \frac{π}{5} ) cos ( \frac{2 π}{5} ) ( cos ( - \frac{4 π}{5} ) + cos ( \frac{12 π}{5} ) )}{2}

cos (\frac{π}{5}) cos (\frac{2 π}{5}) \frac{1}{2} (cos (- \frac{4 π}{5}) + cos (\frac{12 π}{5}))

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot cos ( \frac{π}{5} ) cos ( \frac{2 π}{5} ) ( cos ( - \frac{4 π}{5} ) + cos ( \frac{12 π}{5} ) )}{2}

Multiply:

1 \cdot cos (\frac{π}{5}) = cos (\frac{π}{5})

= \frac{cos ( \frac{π}{5} ) cos ( \frac{2 π}{5} ) ( cos ( - \frac{4 π}{5} ) + cos ( \frac{12 π}{5} ) )}{2}

= \frac{cos ( \frac{π}{5} ) cos ( \frac{2 π}{5} ) ( cos ( - \frac{4 π}{5} ) + cos ( \frac{12 π}{5} ) )}{2}

Use the following property:

cos (- x) = cos (x)

cos (- \frac{4 π}{5}) = cos (\frac{4 π}{5})

= \frac{cos ( \frac{π}{5} ) cos ( \frac{2 π}{5} ) ( cos ( \frac{4 π}{5} ) + cos ( \frac{12 π}{5} ) )}{2}

cos (\frac{12 π}{5}) = cos (\frac{2 π}{5})

cos (\frac{12 π}{5})

Rewrite

\frac{12 π}{5}

2 π + \frac{2 π}{5}

= cos (2 π + \frac{2 π}{5})

Apply the periodicity of

cos

cos (x + 2 π) = cos (x)

cos (2 π + \frac{2 π}{5}) = cos (\frac{2 π}{5})

= cos (\frac{2 π}{5})

= \frac{cos ( \frac{π}{5} ) cos ( \frac{2 π}{5} ) ( cos ( \frac{4 π}{5} ) + cos ( \frac{2 π}{5} ) )}{2}

= \frac{cos ( \frac{π}{5} ) cos ( \frac{2 π}{5} ) ( cos ( \frac{4 π}{5} ) + cos ( \frac{2 π}{5} ) )}{2}

\frac{cos ( \frac{π}{5} ) cos ( \frac{2 π}{5} ) ( cos ( \frac{4 π}{5} ) + cos ( \frac{2 π}{5} ) )}{2} = \frac{cos ( \frac{π}{5} ) cos ( \frac{2 π}{5} ) cos ( \frac{4 π}{5} ) + cos ( \frac{π}{5} ) cos ^{2} ( \frac{2 π}{5} )}{2}

\frac{cos ( \frac{π}{5} ) cos ( \frac{2 π}{5} ) ( cos ( \frac{4 π}{5} ) + cos ( \frac{2 π}{5} ) )}{2}

Expand

cos (\frac{π}{5}) cos (\frac{2 π}{5}) (cos (\frac{4 π}{5}) + cos (\frac{2 π}{5})) : cos (\frac{π}{5}) cos (\frac{2 π}{5}) cos (\frac{4 π}{5}) + cos (\frac{π}{5}) cos^{2} (\frac{2 π}{5})

cos (\frac{π}{5}) cos (\frac{2 π}{5}) (cos (\frac{4 π}{5}) + cos (\frac{2 π}{5}))

Apply the distributive law:

a (b + c) = ab + a c

a = cos (\frac{π}{5}) cos (\frac{2 π}{5}), b = cos (\frac{4 π}{5}), c = cos (\frac{2 π}{5})

= cos (\frac{π}{5}) cos (\frac{2 π}{5}) cos (\frac{4 π}{5}) + cos (\frac{π}{5}) cos (\frac{2 π}{5}) cos (\frac{2 π}{5})

cos (\frac{π}{5}) cos (\frac{2 π}{5}) cos (\frac{2 π}{5}) = cos (\frac{π}{5}) cos^{2} (\frac{2 π}{5})

cos (\frac{π}{5}) cos (\frac{2 π}{5}) cos (\frac{2 π}{5})

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

cos (\frac{2 π}{5}) cos (\frac{2 π}{5}) = cos^{1 + 1} (\frac{2 π}{5})

= cos (\frac{π}{5}) cos^{1 + 1} (\frac{2 π}{5})

Add the numbers:

1 + 1 = 2

= cos (\frac{π}{5}) cos^{2} (\frac{2 π}{5})

= cos (\frac{π}{5}) cos (\frac{2 π}{5}) cos (\frac{4 π}{5}) + cos (\frac{π}{5}) cos^{2} (\frac{2 π}{5})

= \frac{cos ( \frac{π}{5} ) cos ( \frac{2 π}{5} ) cos ( \frac{4 π}{5} ) + cos ( \frac{π}{5} ) cos ^{2} ( \frac{2 π}{5} )}{2}

= \frac{cos ( \frac{π}{5} ) cos ( \frac{2 π}{5} ) cos ( \frac{4 π}{5} ) + cos ( \frac{π}{5} ) cos ^{2} ( \frac{2 π}{5} )}{2}