Is csc^2(θ/2)= 2/(1-cos(θ)) ?

The answer to whether csc^2(θ/2)= 2/(1-cos(θ)) is True

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

prove csc^2(θ/2)= 2/(1-cos(θ))

Solution

prove

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

Solution

True

Solution steps

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

Let:

u = \frac{θ}{2}

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

Prove

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

True

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

Manipulating right side

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Simplify

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

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

Divide the numbers:

\frac{2}{2} = 1

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{1}{( \frac{1}{c s c ( u )} ) ^{2}}

Simplify

\frac{1}{( \frac{1}{c s c ( u )} ) ^{2}}

(\frac{1}{csc ( u )})^{2} = \frac{1}{csc ^{2} ( u )}

(\frac{1}{csc ( u )})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{csc ^{2} ( u )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{csc ^{2} ( u )}

= \frac{1}{\frac{1}{c s c ^{2} ( u )}}

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{csc ^{2} ( u )}{1}

Apply rule

\frac{a}{1} = a

= csc^{2} (u)

csc^{2} (u)

csc^{2} (u)

We showed that the two sides could take the same form

\Rightarrow True

Therefore

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

We showed that the two sides could take the same form

\Rightarrow True

Popular Examples

prove sec(x)cos(x)+tan^2(x)=sec^2(x)prove

sec (x) cos (x) + tan^{2} (x) = sec^{2} (x)

prove (tan^2(θ)+1)/(sec(θ))=sec(θ)prove

\frac{tan ^{2} ( θ ) + 1}{sec ( θ )} = sec (θ)

prove cot^2(θ)+1= 1/(sin^2(θ))prove

cot^{2} (θ) + 1 = \frac{1}{sin ^{2} ( θ )}

prove 3cos(2x+pi/2)-2=6cos^2(x+pi/4)prove

3 cos (2 x + \frac{π}{2}) - 2 = 6 cos^{2} (x + \frac{π}{4})

prove (sin(x))/(cos(x)cot(x^2))=tan(x)prove

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

Frequently Asked Questions (FAQ)

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