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

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

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Popular Trigonometry >

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

Solution

prove

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

Solution

True

Solution steps

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

Let:

u = \frac{x}{2}

csc^{2} (u) = \frac{2}{1 - cos ( 2 u )}

Prove

csc^{2} (u) = \frac{2}{1 - cos ( 2 u )} :

True

csc^{2} (u) = \frac{2}{1 - cos ( 2 u )}

Manipulating right side

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Simplify

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

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

Expand

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)

= \frac{2}{2 sin ^{2} ( u )}

Divide the numbers:

\frac{2}{2} = 1

= \frac{1}{sin ^{2} ( u )}

= \frac{1}{sin ^{2} ( u )}

= \frac{1}{sin ^{2} ( u )}

Rewrite using trig identities

Use the basic trigonometric identity:

sin (x) = \frac{1}{csc ( x )}

\frac{1}{( \frac{1}{c s c ( u )} ) ^{2}}

Simplify

\frac{1}{( \frac{1}{c s c ( u )} ) ^{2}}

(\frac{1}{csc ( u )})^{2} = \frac{1}{csc ^{2} ( u )}

(\frac{1}{csc ( u )})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{csc ^{2} ( u )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{csc ^{2} ( u )}

= \frac{1}{\frac{1}{c s c ^{2} ( u )}}

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{csc ^{2} ( u )}{1}

Apply rule

\frac{a}{1} = a

= csc^{2} (u)

csc^{2} (u)

csc^{2} (u)

We showed that the two sides could take the same form

\Rightarrow True

Therefore

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

We showed that the two sides could take the same form

\Rightarrow True

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