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

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

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

prove

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

True

Solution steps

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

Let:

u = \frac{x}{2}

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

Prove

cot (u) = \frac{sin ( 2 u )}{1 - cos ( 2 u )} :

True

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

Manipulating right side

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

Rewrite using trig identities

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

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

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

Use the Double Angle identity:

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

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

Simplify

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

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

Divide the numbers:

\frac{2}{2} = 1

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

Cancel the common factor:

sin (u)

= \frac{cos ( u )}{sin ( u )}

= \frac{cos ( u )}{sin ( u )}

= \frac{cos ( u )}{sin ( u )}

Use the basic trigonometric identity:

\frac{cos ( x )}{sin ( x )} = cot (x)

= cot (u)

We showed that the two sides could take the same form

\Rightarrow True

Therefore

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

We showed that the two sides could take the same form

\Rightarrow True

cos^{4} (x) - sin^{4} (x) + 1 = 2 cos^{2} (x)

sin (x) cos (x) = \frac{sin ( 2 x )}{2}

- tan (y) + sec (y) = \frac{cos ( y )}{1 + sin ( y )}

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

sin (2 z) = 2 sin (z) cos (z)

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