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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply the fraction rule:

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

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

Cancel the common factor:

cos (x)

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

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

Simplify

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

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= csc (x)

csc (x)

csc (x)

We showed that the two sides could take the same form

\Rightarrow True

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

2 sin^{2} (2 t) + cos (4 t) = 1

\frac{cos ( θ ) + cos ( 3 θ )}{2 cos ( 2 θ )} = cos (θ)

tan^{3} (θ) = tan (θ) sec^{2} (θ) - tan (θ)

sin^{2} (θ) + sin (θ) = 1

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