What is the value of cos^2((9pi)/8)-cos(2*(9pi)/8) ?

The value of cos^2((9pi)/8)-cos(2*(9pi)/8) is (2-sqrt(2))/4

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cos^2((9pi)/8)-cos(2*(9pi)/8)

Solution

cos^{2} (\frac{9 π}{8}) - cos (2 \cdot \frac{9 π}{8})

Solution

\frac{2 - 2}{4}

Decimal Notation

0.14644 \dots

Solution steps

cos^{2} (\frac{9 π}{8}) - cos (2 \cdot \frac{9 π}{8})

Simplify:

2 \cdot \frac{9 π}{8} = \frac{9 π}{4}

2 \cdot \frac{9 π}{8}

Multiply fractions:

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

= \frac{9 π 2}{8}

Multiply the numbers:

9 \cdot 2 = 18

= \frac{18 π}{8}

Cancel the common factor:

2

= \frac{9 π}{4}

= cos^{2} (\frac{9 π}{8}) - cos (\frac{9 π}{4})

cos (\frac{9 π}{4}) = cos (\frac{π}{4})

cos (\frac{9 π}{4})

Rewrite

\frac{9 π}{4}

2 π + \frac{π}{4}

= cos (2 π + \frac{π}{4})

Apply the periodicity of

cos

cos (x + 2 π) = cos (x)

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

= cos (\frac{π}{4})

= cos^{2} (\frac{9 π}{8}) - cos (\frac{π}{4})

Rewrite using trig identities:

cos^{2} (\frac{9 π}{8}) = \frac{1 + cos ( \frac{π}{4} )}{2}

cos^{2} (\frac{9 π}{8})

Use the following identity:

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

Use the Double Angle identity

cos (2 θ) = 2 cos^{2} (θ) - 1

Switch sides

2 cos^{2} (θ) - 1 = cos (2 θ)

Add

1

to both sides

2 sin^{2} (θ) = 1 + cos (2 θ)

Divide both sides by

2

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

= \frac{1 + cos ( 2 \cdot \frac{9 π}{8} )}{2}

Simplify:

2 \cdot \frac{9 π}{8} = \frac{9 π}{4}

2 \cdot \frac{9 π}{8}

Multiply fractions:

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

= \frac{9 π 2}{8}

Multiply the numbers:

9 \cdot 2 = 18

= \frac{18 π}{8}

Cancel the common factor:

2

= \frac{9 π}{4}

= \frac{1 + cos ( \frac{9 π}{4} )}{2}

cos (\frac{9 π}{4}) = cos (\frac{π}{4})

cos (\frac{9 π}{4})

Rewrite

\frac{9 π}{4}

2 π + \frac{π}{4}

= cos (2 π + \frac{π}{4})

Apply the periodicity of

cos

cos (x + 2 π) = cos (x)

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

= cos (\frac{π}{4})

= \frac{1 + cos ( \frac{π}{4} )}{2}

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

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Add similar elements:

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

= \frac{1 - cos ( \frac{π}{4} )}{2}

= \frac{1 - cos ( \frac{π}{4} )}{2}

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{2}{2}

cos (\frac{π}{4})

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

= \frac{1 - \frac{2}{2}}{2}

Simplify

\frac{1 - \frac{2}{2}}{2} : \frac{2 - 2}{4}

\frac{1 - \frac{2}{2}}{2}

Join

1 - \frac{2}{2} : \frac{2 - 1}{2}

1 - \frac{2}{2}

Convert element to fraction:

1 = \frac{1 \cdot 2}{2}

= \frac{1 \cdot 2}{2} - \frac{2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{1 \cdot 2 - 2}{2}

Multiply the numbers:

1 \cdot 2 = 2

= \frac{2 - 2}{2}

Factor

2 - 2 : 2 (2 - 1)

2 - 2

2 = 22

= 22 - 2

Factor out common term

2

= 2 (2 - 1)

= \frac{2 ( 2 - 1 )}{2}

Cancel

\frac{2 ( 2 - 1 )}{2} : \frac{2 - 1}{2}

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

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2 ^{\frac{1}{2}} ( 2 - 1 )}{2}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{2}}}

= \frac{2 - 1}{2 ^{1 - \frac{1}{2}}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{2 - 1}{2 ^{\frac{1}{2}}}

Apply radical rule:

2^{\frac{1}{2}} = 2

= \frac{2 - 1}{2}

= \frac{2 - 1}{2}

= \frac{\frac{2 - 1}{2}}{2}

Apply the fraction rule:

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

= \frac{2 - 1}{2 \cdot 2}

Rationalize

\frac{2 - 1}{2 2} : \frac{2 - 2}{4}

\frac{2 - 1}{2 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{( 2 - 1 ) 2}{2 \cdot 2 2}

(2 - 1) 2 = 2 - 2

(2 - 1) 2

= 2 (2 - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 2, c = 1

= 22 - 2 \cdot 1

= 22 - 1 \cdot 2

Simplify

22 - 1 \cdot 2 : 2 - 2

22 - 1 \cdot 2

Apply radical rule:

a a = a

22 = 2

= 2 - 1 \cdot 2

Multiply:

1 \cdot 2 = 2

= 2 - 2

= 2 - 2

2 \cdot 22 = 4

2 \cdot 22

Apply exponent rule:

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

222 = 2 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2} + \frac{1}{2}}

= 2^{1 + \frac{1}{2} + \frac{1}{2}}

Add similar elements:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

= 2^{1 + 2 \cdot \frac{1}{2}}

2 \cdot \frac{1}{2} = 1

2 \cdot \frac{1}{2}

Multiply fractions:

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

= \frac{1 \cdot 2}{2}

Cancel the common factor:

2

= 1

= 2^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2^{2}

2^{2} = 4

= 4

= \frac{2 - 2}{4}

= \frac{2 - 2}{4}

= \frac{2 - 2}{4}

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Frequently Asked Questions (FAQ)

What is the value of cos^2((9pi)/8)-cos(2*(9pi)/8) ?
The value of cos^2((9pi)/8)-cos(2*(9pi)/8) is (2-sqrt(2))/4