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

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

prove

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

True

Solution steps

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

Manipulating right side

cot^{2} (x)

Express with sin, cos

cot^{2} (x)

Use the basic trigonometric identity:

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

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

Simplify

(\frac{cos ( x )}{sin ( x )})^{2} : \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

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

Apply exponent rule:

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

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

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

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

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

Use the Pythagorean identity:

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

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

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

Use the Double Angle identity:

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

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

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

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

Group like terms

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

Add similar elements:

cos^{2} (x) - cos^{2} (x) = 0

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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

sin (2 θ) + 2 sin (θ) - cos (θ) = 2

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

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

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

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