Is cos^2(((12pi))/2)=cos^2(12pi) ?

The answer to whether cos^2(((12pi))/2)=cos^2(12pi) is True

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prove cos^2(((12pi))/2)=cos^2(12pi)

prove

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

True

Solution steps

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

Manipulating left side

cos^{2} (\frac{12 π}{2})

Simplify

cos^{2} (\frac{12 π}{2}) : 1

cos^{2} (\frac{12 π}{2})

Divide the numbers:

\frac{12}{2} = 6

= cos^{2} (6 π)

cos (6 π) = 1

cos (6 π)

cos (6 π) = cos (0)

cos (6 π)

Rewrite

6 π

2 π \cdot 3 + 0

= cos (2 π 3 + 0)

Apply the periodicity of

cos

cos (x + 2 π \cdot k) = cos (x)

cos (2 π \cdot 3 + 0) = cos (0)

= cos (0)

= cos (0)

Use the following trivial identity:

cos (0) = 1

cos (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}

= 1

= 1

= 1^{2}

Apply rule

1^{a} = 1

= 1

= 1

Manipulating right side

cos^{2} (12 π)