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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

prove 1+cos(x)=(sin^2(x))/(1-cos(x))

prove

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

True

Solution steps

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

Manipulating right side

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Factor

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

1 - cos^{2} (x)

Factor out common term

- 1

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

Factor

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

cos^{2} (x) - 1

Rewrite

1

1^{2}

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

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

= (cos (x) + 1) (cos (x) - 1)

= - (cos (x) + 1) (cos (x) - 1)

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

Cancel

- \frac{( cos ( x ) + 1 ) ( cos ( x ) - 1 )}{1 - cos ( x )} : cos (x) + 1

- \frac{( cos ( x ) + 1 ) ( cos ( x ) - 1 )}{1 - cos ( x )}

- cos (x) + 1 = - (cos (x) - 1)

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

Refine

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

Cancel the common factor:

cos (x) - 1

= cos (x) + 1

= cos (x) + 1

= cos (x) + 1

= cos (x) + 1

= 1 + cos (x)

We showed that the two sides could take the same form

\Rightarrow True

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

sec^{2} (θ) csc^{2} (θ) = sec^{2} (θ) + csc^{2} (θ)

sin (2 x) - tan (x) cos (2 x) = tan (x)

sin (θ) tan (θ) + cos (θ) = \frac{1}{cos ( θ )}

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

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