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

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

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prove 1-cos(x)=2sin^2(x/2)

prove

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

True

Solution steps

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

Let:

u = \frac{x}{2}

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

Prove

1 - cos (2 u) = 2 sin^{2} (u) :

True

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

Manipulating left side

1 - cos (2 u)

Rewrite using trig identities

1 - cos (2 u)

Use the Double Angle identity:

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

= 1 - (1 - 2 sin^{2} (u))

Simplify

1 - (1 - 2 sin^{2} (u)) : 2 sin^{2} (u)

1 - (1 - 2 sin^{2} (u))

- (1 - 2 sin^{2} (u)) : - 1 + 2 sin^{2} (u)

- (1 - 2 sin^{2} (u))

Distribute parentheses

= - (1) - (- 2 sin^{2} (u))

Apply minus-plus rules

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

= - 1 + 2 sin^{2} (u)

= 1 - 1 + 2 sin^{2} (u)

1 - 1 = 0

= 2 sin^{2} (u)

= 2 sin^{2} (u)

= 2 sin^{2} (u)

We showed that the two sides could take the same form

\Rightarrow True

Therefore

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

We showed that the two sides could take the same form

\Rightarrow True

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csc^{6} (x) - cot^{6} (x) = 1 + 3 csc^{2} (x) cot^{2} (x)

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