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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

prove

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

True

Solution steps

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

Let:

u = \frac{x}{2}

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

Prove

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

True

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

Manipulating left 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

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

We showed that the two sides could take the same form

\Rightarrow True

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

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

1 = (sec (2 x))^{2} - (tan (2 x))^{2}

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

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

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