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

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

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prove (cot^2(x))/(cos^2(x))=csc^2(x)

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Simplify

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

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

Apply the fraction rule:

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

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

Cancel the common factor:

cos^{2} (x)

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= csc^{2} (x)

csc^{2} (x)

csc^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

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

sec (x) = arccos (x)

tan (θ) + sin (θ) = 4 (1 + cos (θ))

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

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

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