Is csc(u)-cot(u)=(sin(u))/(1+cos(u)) ?

The answer to whether csc(u)-cot(u)=(sin(u))/(1+cos(u)) is True

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

prove csc(u)-cot(u)=(sin(u))/(1+cos(u))

prove

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

True

Solution steps

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

Manipulating left side

csc (u) - cot (u)

Express with sin, cos

- cot (u) + csc (u)

Use the basic trigonometric identity:

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

= - \frac{cos ( u )}{sin ( u )} + csc (u)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply rule

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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

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

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

Rewrite using trig identities

Multiply by

\frac{1 + cos ( u )}{1 + cos ( u )}

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

Expand

(1 + cos (u)) (1 - cos (u)) : 1 - cos^{2} (u)

(1 + cos (u)) (1 - cos (u))

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = 1, b = cos (u)

= 1^{2} - cos^{2} (u)

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - cos^{2} (u)

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

Use the Pythagorean identity:

1 = cos^{2} (u) + sin^{2} (u)

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

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

Cancel the common factor:

sin (u)

= \frac{sin ( u )}{1 + cos ( u )}

= \frac{sin ( u )}{1 + cos ( u )}

We showed that the two sides could take the same form

\Rightarrow True

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

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

sin (2 x) = sin (x + x)

8 cos^{4} (x) - 8 cos^{2} (x) + 1 = cos (4 x)

\frac{cos ( 2 x ) - cos ( 4 x )}{sin ( 2 x ) sin ( 4 x )} = tan (x)

Is csc(u)-cot(u)=(sin(u))/(1+cos(u)) ?
The answer to whether csc(u)-cot(u)=(sin(u))/(1+cos(u)) is True