Is (cot^2(x))/(csc^2(x))=cos^2(x) ?

The answer to whether (cot^2(x))/(csc^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))/(csc^2(x))=cos^2(x)

prove

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

True

Solution steps

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

Manipulating left side

\frac{cot ^{2} ( x )}{csc ^{2} ( x )}

Express with sin, cos

\frac{cot ^{2} ( x )}{csc ^{2} ( x )}

Use the basic trigonometric identity:

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

= \frac{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}{csc ^{2} ( x )}

Use the basic trigonometric identity:

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

= \frac{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}{( \frac{1}{s i n ( x )} ) ^{2}}

Simplify

\frac{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}{( \frac{1}{s i n ( x )} ) ^{2}} : cos^{2} (x)

\frac{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}{( \frac{1}{s i n ( x )} ) ^{2}}

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

= \frac{( \frac{c o s ( x )}{s i n ( x )} ) ^{2}}{\frac{1}{s i n ^{2} ( x )}}

(\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{\frac{c o s ^{2} ( x )}{s i n ^{2} ( x )}}{\frac{1}{s i n ^{2} ( x )}}

Divide fractions:

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

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

Refine

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

Cancel the common factor:

sin^{2} (x)

= cos^{2} (x)

= cos^{2} (x)

= cos^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

(cos (x) - sin (x)) 2 = 1 - 2 sin (x) cos (x)

cos (x) (csc^{2} (x)) = cot (x) csc (x)

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

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

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

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