Is 2cos(2(pi/4))=sqrt(16cos(2(pi/4))) ?

The answer to whether 2cos(2(pi/4))=sqrt(16cos(2(pi/4))) is True

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prove 2cos(2(pi/4))=sqrt(16cos(2(pi/4)))

Solution

prove

2 cos (2 (\frac{π}{4})) = 16 cos (2 (\frac{π}{4}))

Solution

True

Solution steps

2 cos (2 \cdot \frac{π}{4}) = 16 cos (2 \cdot \frac{π}{4})

Manipulating left side

2 cos (2 \frac{π}{4})

Simplify

2 cos (2 \frac{π}{4}) : 0

2 cos (2 \cdot \frac{π}{4})

Multiply

2 \cdot \frac{π}{4} : \frac{π}{2}

2 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 2}{4}

Cancel the common factor:

2

= \frac{π}{2}

= 2 cos (\frac{π}{2})

Simplify

cos (\frac{π}{2}) : 0

cos (\frac{π}{2})

Use the following trivial identity:

cos (\frac{π}{2}) = 0

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}

= 0

= 2 \cdot 0

Apply rule

0 \cdot a = 0

= 0

= 0

Manipulating right side

16 cos (2 \frac{π}{4})

Simplify

16 cos (2 \frac{π}{4}) : 0

16 cos (2 \cdot \frac{π}{4})

Multiply

2 \cdot \frac{π}{4} : \frac{π}{2}

2 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 2}{4}

Cancel the common factor:

2

= \frac{π}{2}

= 16 cos (\frac{π}{2})

Simplify

cos (\frac{π}{2}) : 0

cos (\frac{π}{2})

Use the following trivial identity:

cos (\frac{π}{2}) = 0

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}

= 0

= 16 \cdot 0

Simplify

16 \cdot 0

Apply rule

0 \cdot a = 0

= 0

Apply rule

0 = 0

= 0

= 0

= 0

We showed that the two sides could take the same form

\Rightarrow True

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Is 2cos(2(pi/4))=sqrt(16cos(2(pi/4))) ?
The answer to whether 2cos(2(pi/4))=sqrt(16cos(2(pi/4))) is True