Is (cot(θ))/(cos(θ))=csc(θ) ?

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

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prove (cot(θ))/(cos(θ))=csc(θ)

prove

\frac{cot ( θ )}{cos ( θ )} = csc (θ)

True

Solution steps

\frac{cot ( θ )}{cos ( θ )} = csc (θ)

Manipulating left side

\frac{cot ( θ )}{cos ( θ )}

Express with sin, cos

\frac{cot ( θ )}{cos ( θ )}

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply the fraction rule:

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

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

Cancel the common factor:

cos (θ)

= \frac{1}{sin ( θ )}

= \frac{1}{sin ( θ )}

= \frac{1}{sin ( θ )}

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{1}{\frac{1}{c s c ( θ )}}

Simplify

\frac{1}{\frac{1}{c s c ( θ )}}

Apply the fraction rule:

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

= \frac{csc ( θ )}{1}

Apply rule

\frac{a}{1} = a

= csc (θ)

csc (θ)

csc (θ)

We showed that the two sides could take the same form

\Rightarrow True

cos^{2} (2 x) = \frac{1}{2} (1 + cos (4 x))

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

sin (4 x) = 8 sin (x) cos (3 x) - 4 sin (x) cos (x)

\frac{cot ( β )}{csc ( β )} = cos (β)

\frac{1 + cos ( w )}{sin ( w )} = \frac{sin ( w )}{1 - cos ( w )}

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