Is 1-cos(pi/n)=2sin^2(pi/(2n)) ?

The answer to whether 1-cos(pi/n)=2sin^2(pi/(2n)) is True

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prove 1-cos(pi/n)=2sin^2(pi/(2n))

prove

1 - cos (\frac{π}{n}) = 2 sin^{2} (\frac{π}{2 n})

True

Solution steps

1 - cos (\frac{π}{n}) = 2 sin^{2} (\frac{π}{2 n})

Manipulating right side

2 sin^{2} (\frac{π}{2 n})

Rewrite using trig identities

2 sin^{2} (\frac{π}{2 n})

Use the Double Angle identity:

1 - 2 sin^{2} (x) = cos (2 x)

- 2 sin^{2} (x) = cos (2 x) - 1

= - (cos (2 \cdot \frac{π}{2 n}) - 1)

Simplify

- (cos (2 \cdot \frac{π}{2 n}) - 1) : - cos (\frac{π}{n}) + 1

- (cos (2 \cdot \frac{π}{2 n}) - 1)

Multiply

2 \cdot \frac{π}{2 n} : \frac{π}{n}

2 \cdot \frac{π}{2 n}

Multiply fractions:

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

= \frac{π 2}{2 n}

Cancel the common factor:

2

= \frac{π}{n}

= - (cos (\frac{π}{n}) - 1)

Distribute parentheses

= - (cos (\frac{π}{n})) - (- 1)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - cos (\frac{π}{n}) + 1

= - cos (\frac{π}{n}) + 1

= - cos (\frac{π}{n}) + 1

= 1 - cos (\frac{π}{n})

We showed that the two sides could take the same form

\Rightarrow True

Is 1-cos(pi/n)=2sin^2(pi/(2n)) ?
The answer to whether 1-cos(pi/n)=2sin^2(pi/(2n)) is True