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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

prove

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

True

Solution steps

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

Manipulating right side

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Use the Double Angle identity:

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

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

Simplify

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

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

Expand

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

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

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

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

Distribute parentheses

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

Apply minus-plus rules

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

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

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

1 - 1 = 0

= 2 sin^{2} (y)

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

Divide the numbers:

\frac{2}{2} = 1

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

Cancel the common factor:

sin (y)

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

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

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

Use the basic trigonometric identity:

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

= cot (y)

We showed that the two sides could take the same form

\Rightarrow True

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