Is cot(θ/2)=(sin(θ))/(1-cos(θ)) ?

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

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prove cot(θ/2)=(sin(θ))/(1-cos(θ))

prove

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

True

Solution steps

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

Let:

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

We showed that the two sides could take the same form

\Rightarrow True

sin (2 θ) = cot (θ) (1 - cos (2 θ))

\frac{cos ( 6 0 ^{\circ} )}{sin ( 6 0 ^{\circ} )} = cot (6 0^{\circ})

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

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

sin (θ + \frac{3 π}{2}) = - cos (θ)

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