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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

Multiply fractions:

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

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Cancel the common factor:

sin^{2} (x)

= cos^{2} (x)

= cos^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

sec^{2} (θ) = 1 + tan^{2} (θ)

1 + tan^{2} (3 0^{\circ}) = sec^{2} (3 0^{\circ})

(sin (x))^{2} - (sec (x)) (sin (x))^{3} = - 1

sin (2 x) (tan (x) + cot (x)) = 2

sec^{4} (x) - tan^{4} (x) = 2 sec^{2} (x) - 1

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