Is (cot(2θ))/(cos(2θ))=csc(2θ) ?

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

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prove (cot(2θ))/(cos(2θ))=csc(2θ)

prove

\frac{cot ( 2 θ )}{cos ( 2 θ )} = csc (2 θ)

True

Solution steps

\frac{cot ( 2 θ )}{cos ( 2 θ )} = csc (2 θ)

Manipulating left side

\frac{cot ( 2 θ )}{cos ( 2 θ )}

Express with sin, cos

\frac{cot ( 2 θ )}{cos ( 2 θ )}

Use the basic trigonometric identity:

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

= \frac{\frac{c o s ( 2 θ )}{s i n ( 2 θ )}}{cos ( 2 θ )}

Simplify

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

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

Apply the fraction rule:

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

= \frac{cos ( 2 θ )}{sin ( 2 θ ) cos ( 2 θ )}

Cancel the common factor:

cos (2 θ)

= \frac{1}{sin ( 2 θ )}

= \frac{1}{sin ( 2 θ )}

= \frac{1}{sin ( 2 θ )}

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{1}{\frac{1}{c s c ( 2 θ )}}

Simplify

\frac{1}{\frac{1}{c s c ( 2 θ )}}

Apply the fraction rule:

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

= \frac{csc ( 2 θ )}{1}

Apply rule

\frac{a}{1} = a

= csc (2 θ)

csc (2 θ)

csc (2 θ)

We showed that the two sides could take the same form

\Rightarrow True

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